COO-2262-5
MITNE-171
FINITE ELEMENT SYNTHESIS METHOD
by
Shi-tien Yang, Allan F. Henry
May 1975
D artment of Nuclear Engineering
A4ssachusetts Institute of Technology
Cambridge, Massachusetts
I
AEC Research and Development Report
Contract AT(1 1-1)-2262
U.S. Atomic Energy Commission
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
DEPARTMENT OF NUCLEAR ENGINEERING
Cambridge,
Massachusetts 02139
FINITE ELEMENT SYNTHESIS METHOD
by
Shi-tien Yang,
Allan F. Henry
May 1975
COO - 2262 - 5
MITNE - 171
AEC Research and Development Report
Contract AT(11-1)2262
U.S. Atomic Energy Commission
FINITE ELEMENT SYNTHESIS METHOD
by
Shi-tien Yang
Submitted to the Department of Nuclear Engineering on May 2, 1975 in
partial fulfillment of the requirements for the degree of Doctor of
Philosophy
ABSTRACT
A coarse mesh approximation method is presented for the solution
of the two-dimensional spatial neutron flux in multigroup diffusion
theory. This so-called finite element synthesis method is consistent in
that it is systematically derived as an extension of the finite element
method by utilizing general variational techniques. Detailed subassembly
solutions, found by imposing zero current boundary conditions over the
surface of each subassembly, are modified by piecewise continuous
Hermite polynomials of the finite element method and used directly in
trial function forms.
The finite element synthesis method differs substantially from the
finite element method in which homogeneous nuclear constants, homogenized
by flux weighting with detailed subassembly solutions, are used. However,
both schemes become equivalent when the subassemblies themselves are
homogeneous.
The application of this method to some representative one-dimensional PWR configurations has been shown to be successful. However, the
extension of two-dimensional problems is not as straightforward as it
might appear to be, because there are flux discontinuities at subassembly
boundaries. Thus, some additional terms, representing the contribution
of the flux (and current) discontinuities, has to be added to the
difference equations of the approximation method.
Two-dimensional, two-group numerical calculations using representative nuclear material constants for fuel, absorber and moderator and
18 cm x 18 cm subassemblies were performed using entire subassemblies as
coarse mesh regions. The results indicate that the finite element synthesis method can yield accurate criticality predictions and detailed flux
shapes throughout a core composed of heterogeneous subassemblies.
Thesis Supervisor:
Thesis Reader:
Titles:
Allan F. Henry
Kent F. Hansen
Professors of Nuclear Engineering
2
ACKNOWLEDGEMENTS
I would like to thank and express my deep appreciation and
sincerest gratitude to Prof. Allan F. Henry, my thesis supervisor,
for the constant guidance, consultation and encouragement provided
during the course of this study.
Appreciation must also be extended
to my thesis reader, Prof. Kent F. Hansen, for his assistance in
reviewing the final manuscript and providing many helpful discussions.
The work was performed under USAEC Contract AT(ll-1)2262,
with the computations performed at the M.I.T. Information Processing
Center and the Laboratory of Nuclear Science.
Special appreciation is extended to my wife, Yu-fen.
Without
her encouragement and understanding, this work would be impossible.
Finally, special thanks are expressed to Mrs. Stephanie Demeris
who typed this thesis with skill and patience.
3
TABLE OF CONTENTS
Page
Abstract
2
Acknowledgements
3
List of Figures
7
List of Tables
10
Chapter 1.
Introduction
12
i.1
Introduction
12
1.2
The Time-Independent, Multigroup
Diffusion Theory Equations
16
Hermite Piecewise Polynomials and
Basis Functions
20
organization of the Thesis
25
Variational Principles in Neutron
Diffusion Problems
26
1.3
1.4
Chapter 2.
2.1
Variational Principles in Diffusion
Theory
28
2.2
Discontinuous Trial Functions
31
2.3
Fick's Law as a Consequence of
Variational Principle
35
2.3.1
2.3.2
2.3.3
The Linear Basis Functions
Approximation
36
The Cubic Hermite Basis
Functions Approximation
47
Summary of Section 2.3
56
4
Page
Investigation of Finite Element Method
Using Discontinuous Current Trial Functions
58
3.1
Derivation of Difference Equations in
1-D
59
3.2
Derivation of Difference Equations in
2-D
61
3.3
Numerical Method
67
3.4
Numerical Results
73
Finite Element Synthesis Approximation
Method in Neutron Diffusion Problems
89
Derivation of the Difference Equations
91
Chapte r 3.
Chapte r 4.
4.1
4.1.1
4.1.2
Linear Basis Function
Approximation in 1-D
91
Bi-linear Basis Function
Approximation in 2-D
92
4.2
Calculational and Programming Techniques
103
4.3
Numerical Results
107
4.3.1
Case 1
--
One Group
107
4.3.2
Case 2
--
One Group
117
4.3.3
Case 3
--
Two Groups
124
Conclusions and Recommendations
147
5.1
Conclusions of the Study
147
5.2
Recommendations for Future Study
149
Chapt er 5.
Bibliography
151
Biographical Note
156
Appendix A. Table of Symbols
157
Appendix B. Inner Products for Hermite Basis Functions
161
5
Page
Difference Equation Coefficients
Resulting from Use of the Finite
Element Approximation Methods in
Chapter 2
164
C.1
Coefficients of the Linear Finite Element
Method Equations
164
C.2
Coefficients of the Cubic Hermite Finite
Element Method Equations
165
Difference Equations Resulting from Use
of Cubic Finite Element Approximations
with Discontinuous Fick's Law Current
Trial Functions
168
D.1
Difference Equations in 1-D
168
D.2
Difference Equations in 2-D
Appendix C.
Appendix D.
Appendix E.
Appendix F.
Difference Equation Coefficients
Resulting from Use of the Finite Element
Synthesis Approximation Method in
2-D
Listings of the Computer Programs
F.1
LlNl
F.2
LlN2
F.3
DOB1
F.4
DOB2
F.5
MAN1
Fi,6
MAN2
170
192
206
207
210
214
222
235
247
6
LIST OF FIGURES
PAge
No.
24
1.1
Linear and Cubic Hermite Basis Functions
2.1
The Flux Trial Functions of Linear
Finite Element Approximation
2.2
Matrix Form of the Linear Finite
Element Method Approximation
2.3
The Linear Current Trial Functions
46
2.4
The Quadratic Current Trial Functions
46
2.5
The Cubic Hermite Basis Functions
Approximation
49
2.6
Matrix Form of the Cubic Hermite Finite
Element Method Approximation
52
62
3.1
The Subdivisions of Rectangular Problem
Domain
3.2
P
Solution Scheme of /A P =
Using Source Iterative Method
3.3
Reactor Configuration for Example 3.1
76
3.4
Reactor Configuration for Example 3.2
80
3.5
Reactor Configuration for Example 3.3
3.6
Flux Shapes:
3.7
Reactor Configuration for Example 3.4
3.8
Thermal Flux Shapes:
4.1
A 25 Subassembly Reactor Configuration
and its 2 Different Types of Subassemblies
Case 1
4.2
85
Example 3.3
Mesh Geometry in
1 and 2
88
Example 3.4
a Subassembly -
7
87
Cases
-
Page
No.
Detailed Flux Solutions at y = 0 cm Case 1
112
Detailed Flux Solutions at y = 4 cm Case 1
113
4.5
Flux at Y = 12 cm - Case 1
114
4.6
Flux at Y = 20 cm - Case 1
115
4.7
Flux at Y = 24 cm - Case 1
116
4.8
A 25 Subassembly Symmetric Reactor
Configuration and Its 2 Different Types
of Subassemblies - Case 2
118
Detailed Flux Solutions for Subassembly 1 - Case 2
120
4.3
4.4
4.9
4.10
Flux at Y
=
12 cm - Case 2
121
4.11
Flux at Y
=
20 cm - Case 2
122
4.12
Fluxat Y
8 cm - Case 2
123
4.13
A 49 Subassembly Symmetric Reactor
Configuration and its 3 Different
Types of Subassemblies - Case 3
126
128
4.14
Detailed Fast Fluxes for Subassemblies
Case 3
4.15
Detailed Thermal Fluxes for Subassemblies - Case 3
129
Detailed Fast Adjoint Fluxes for Subassemblies - Case 3
130
Detailed Thermal Adjoint Fluxes for
Subassemblies - Case 3
131
4.16
4.17
-
4.18
Fast Flux at Y
=
27 cm - Case 3
132
4.19
Fast Flux at Y
=
63 cm - Case 3
133
4.20
Thermal Flux at Y = 27 cm - Case 3
8
134
No.
.age
4.21
Thermal Flux at Y = 63 cm
4.22
Flux Shapes for the Water Subassemblies
at the Left Side Used in Modified
4.23
-
Case 3
135
Calculation - Case 3
138
Flux Shapes for the Top-left Corner Water
Subassembly Used in Modified Calculation Case 3
139
4.24
Fast Flux at Y
=
27 cm - Case.3, Modified
141
4.25
Fast Flux at Y
=
63 cm - Case 3, Modified
142
4.26
Thermal Flux at Y
=
27 cm - Case 3,
143
Modified
4.27
Thermal Flux at Y = 63 cm - Case
Modified
9
3,
144
LIST OF TABLES
Page
No.
Matrix Order N of the Bi-cubic Hermite
Basis Functions Approximations with
Discontinuous Current Trial Functions
68
3.2
Two-group Nuclcar Constants
74
3.3
Results of One-dimensional, Two-group,
One-region Problem: Example 3.1
76
One-dimensional, Two-group, Two-region
Eigenvalue Problem: Example 3.2
80
Eigenvalue of a Two-dimensional, Onegroup, One-region Projlem: Example 3.3
84
Eigenvalue of a Two-dimensional, Twogroup, Two-region Problem: Example 3.4
87
3.1
3.4
3.5
3.6
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Matrix Order N of the Bi-linear Finite
Element Synthesis Approximations as a
Function of the Imposed Boundary Conditions
102
One-group Nuclear Constants of Subassemblies of Figure 4.1 - Case 1
110
Homogenized Subassembly One-group
Nuclear Constants - Case 1
110
One-group Nuclear Constants of Subassemblies of Figure 4.8 - Case 2
119
Homogenized Subassembly One-group
Nuclear Constants - Case 2
119
Two-group Nuclear Constants of Three
Different Materials in Subassemblies
of Figure 4.13 - Case 3
127
Homogenized Subassembly Two-group
Nuclear Constants - Case 3
127
10
Page
No.
4.8
4.9
One-group Eigenvalue A Obtained from
Different Methods - Cases 1 and 2
146
Two-group Eigenvalue A Obtained from
Different Methods - Case 3
146
11
CHAPTER 1
INTRODUCTION
1.1
Introduction
This thesis is concerned with the numerical solutions for static
neutron diffusion problems using finite element piecewise polynomials
combined with synthesis techniques in space variables.
The study of the solutions for the neutron diffusion problem
is important, both in reactor economics and reactor safety.
One
major concern of the reactor physicist is that the prediction of
the behavior of a reactor after any foreseeable nuclear accident.
A detailed safety analysis can only be obtained if all the physical
processes occuring within the reactor can be fully understood and
related to each other.
Since all these processes can be shown to
be independent on the neutron flux distribution throughout the
reactor, a detailed and accurate solution of the spatial neutron
flux is vital [1].
A sufficiently detailed description of the physical process
occurring within a nuclear reactor is the Boltzmann neutron transport
equation [2]-[3].
It is essentially a neutron balance equation in
any point, any angle within a reactor system at any time and is
very difficult to solve.
The P-1 and diffusion theory approximation
[4] greatly simplifies the transport equation into more tractable
forms and has been found to give an adequate approximation for the
12
flux distribution for most large-core reactors.
Because of the complexity of reactor geometries and nuclear
cross sections,
numerical methods have been widely used to solve
the neutron diffusion problems and have been shown to be more powerful than analytical methods.
The most widely used method is
the
finite difference method [5]-[6], which is quite simple in principle
but which requires relatively small meshes and hence a large number
of unknowns.
Thus, if hundreds of thousands of mesh points are
desired, it is extremely expensive to obtain the solution today.
For this reason, finite difference methods have generally been
limited to kinetic problems involving only a few thousand mesh
points or to relatively coarse mesh three-dimensional problems, and
alternate methods have been developed which require determination
of a smaller number of unknowns and yet which can be applied to
complex multidimensional problems.
During the past fifteen years or so a class of approximation
methods known as "synthesis methods" have been developed.
The
theoretical basis of modern flux synthesis methods was introduced
into reactor physics by Selengut [7] within the context of variational analysis.
Calame and Federighi [8] and Kaplan [9]-[10] wer
among the first to exploit these methods for synthesizing the
spectral and spatial dependences, respectively, of the neutron flux
distribution.
Other workers [1l]-[19] have subsequently extended
and tested these techniques until today there exists a sizable
13
literature on these methods.
The central idea of the synthesis methods is to express the
soltuion in terms of a small number of functions chosen to represent
various transient states of the problem.
The advantage of this
method is that the expansion functions may be based on the knowledge
of a particular system.
However, the selection of proper expansion
functions for various systems is difficult particularly if feedback
effects of a priori, unknown magnitude are involved.
Poor selection
of expansion functions can misrepresent the solution to an extent
that is not determinable in a systematic way.
Therefore, there is
a need to improve on the synthesis procedure, particularly for
space dependent kinetic problems for complex geometrical systems.
Another computational technique which has been most highly
developed in structural mechanics [20]-[21] and fluid flow [22]-[23]
is the finite element method.
Its application to the neutron
diffusion equation made a few years ago has been shown to be quite
successful [24]-[26].
Briefly, it is a process by which, given the
defining equation of a problem, one seeks to discretize the equation
by dividing the problem domain into a substantial number of subdomains, referred to as elements.
Interpolation functions (usually
piecewise polynomials) are formulated within each element in
terms of parameters associated with nodes on the element boundaries.
Then these nodal parameters are related by the use of some continuity
conditions across the interfaces of adjacent elements.
14
The
variational method or weighted residual method is then applied to
yield a set of simultaneous algebraic equations for the nodal
parameters.
Finally, the approximate solution to the problem is
obtained by the use of a computer.
The advantage of the finite element method is that for a
given degree of accuracy, it requires a smaller number of mesh
points and thus hopefully less computational time relative to the
conventional finite difference method.
Also, (in distinction to
the synthesis method) an error analysis can be performed to find
the error bounds of the approximate solution [27]-[29].
However,
when this coarse mesh method is applied to reactor systems with very
complex geometrical complications,
in
the important flux dips and peaks
small control rods and water holes,
respectively,
cannot be
accurately predicted unless a considerable number of meshes are
placed in
Thus,
these regions.
the problem comes back to that of
reducing the number of unknowns while retaining the accuracy of the
flux distribution in highly heterogeneous regions.
The idea of combining finite element basis functions with the
synthesis technique in
first
solving the neutron diffusion problems was
advanced by Bailey and Henry (30].
They used both linear
and cubic Hermite piecewise polynomials multiplied into detailed
subassembly solutions to describe the neturon flux and current
and then used a variational principle to derive the desired set
of difference equations.
This method was applied to some
15
representative one-dimensional PWR configurations, consisting of
many cells and showed good results compared with those of coarse
mesh finite element methods.
one-dimensional,
However, because the problems were
there were no continuities at cell boundaries.
Thus it is not possible on the basis of this preliminary work to
provide a reliable evaluation of the potential of the idea.
The purpose of this thesis is to extend the aforementioned
finite element-synthesis scheme to two-dimensional problems by
using a variational technique.
Such an extension is not as
straightforward as it might appear to be since, in the two
dimensional case, there are flux discontinuities at cell boundaries.
For example,
consider two adjacent subassemblies in
a square lattice.
If one cell contains a cross-shaped control rod and the other has
its control rod out resulting in a large cross-shaped water hole,
the flux found by stitching two, zero-current subassembly solutions
together will be discontinuous at the interface.
Since potential
errors in overall neutron balance and detailed flux shape are the
matters of chief concern in extending to two dimensions, the thesis
deals only with the spatial approximation for static cases.
Extension to time dependent problems is left for future study.
1.2
The Time-independent, Multigroup Diffusion Theory Equations
In this section, the time-independent, energy-discretized
multigroup P-1 approximation to the Boltzmann neutron transport
16
equation is introduced.
[31] and elsewhere [4],
The derivation can be found in Glasstone
[32].
The standard form of the P-1 equations for each energy group
g is as follows:
j
+ D (r)V# (r)
(r)
=
(1.la)
0
G
(r)
_-
9
+ Ig(r)# (r) -
()
(rO
g'Ig
G
G
where the group index g runs from the highest energy group, 1, to
the lowest energy group, G.
The symbols and notations used through-
out this thesis are summarized in Appendix A.
The net current
vector Jg(r) may be eliminated via Fick's Law,
Equation 1.la, to
obtain the multigroup diffusion equations:
G
-
D (r)V
(r) + 1 (1)0 (1) -
,(r) 0 ,(1) =
I)
g
g-g
1 C
l gg
g
g()
Both Equations 1.1 and 1.2 can be written in matrix form as
17
(.2)
J(r) +JD(r)V?(r)
V.J (r)
+ [ IM(r)
=
-
(1.3a)
0
r)]@(r)
(1. 3b)
IF(r)@(r)
=
and
-V- ID(r) V(r) + [IM(r)
-
where ID(r), M(r),
respectively,
E(r)]G(r)
U(r),
r)
(1.4)
and F(r) are GxG matrices
defined by
JD(r) = Diag[Dl(r), D 2 (r),---, DG(r)]
(1.5a)
14(r) = Diag[jl(r) , 2
(1.5b)
T(r)
=
G -r) ]
0
~21(r
0--------
-12G
['vflg(-r),
(1.5c)
0
-IG1(r)
IF(r) = Xl
r
N(f2 -r* ~~-
fG(r)]
(1.5d)
X2
XG
and J(r) is
the group current vector
J(r) = Col[l (r), J 2 (r),...,jG(r)]
18
(1.5e)
and
(r)
is
the group flux vector
@()=Col[#i
'''Gr](1.5f)
$2
),
It is also convenient to define the GxG group absorption,
scattering the production matrix A(r)
(1.5g)
F (r)
-
A(r) = IM(r) - T (r)
so that Equations 1.1 and 1.2 may be written simply as
J(r) + D(r)VY(r)
V-J(r) + A(r)@(r)
(1.6a)
0
=
=
(1.6b)
0
and
(1.7)
-V. ID(r)?@(r) + A(r)cD(r) = 0
These forms of the group diffusion equations will be
respectively.
The boundary conditions on @(r)
used throughout this thesis.
accompanying these equations are of the homogeneous Dirichlet or
Neumann type [33], namely
.
a
1
)
an
+4 a2
a2
(r(r
)D =r 0
-=*b
if a1 = 0,
then
if a2 = 0,
then
()
=
0,
Dirichlet type
(1.8a)
Neumann type
(1.8b)
a =b
where r
- 0,
denotes the spatial vector on the boundary surface and n
is the unit vector normal to the boundary surface.
19
The continuity conditions imposed on the solution require
that <b(r) be continuous throughout the problem domain and the normal
component of J(r) be continuous across internal interfaces separating
two different material media.
1.3
Hermite Piecewise Polynomials and Basis Functions
Piecewise polynomials are polynomials defined only over
subregions of the problem domain, rather than over the entire domain.
The piecewise constants used in finite difference approximation can
be thought of as a- special case of the piecewise polynomials.
Piecewise polynomials yield high accuracy for approximations of
functions and their derivatives.
Furthermore,
besides the simplicity
of differentiation and integration for practical computations, they
have the following convenient features:
1)
Piecewise polynomials permit flexibility in imposing the
continuity or jump conditions at the joints of subregions.
Also, boundary conditions can be easily imposed.
2)
They provide local approximations and are thus suitable
for approximating the physical behavior of the problems
in which variations occur locally.
example, the reactor system is
In our case, for
characterized by local
variations of the materials and neutron cross sections,
and the use of a few piecewise polynomials is more
convenient than use of polynomials defined over the
20
entire reactor.
3)
The values of functions and their derivatives can be
directly incorporated into the expansion coefficients
proper piecewise polynomial basis functions are used.
if
There are many varieties of piecewise polynomials used in
numerical analysis [34]-[36].
In this section we limit the dis-
cussion to Hermite interpolating functions because of their simplicity.
For the convenience of the discussion, we shall use
dimensionless variables.
Let us divide a one-dimensional problem domain (O,Z] into
K adjoining regions.
Each region k is bounded by nodes zk and
zk+l and has width hk = zk+l - zk.
It
different.
is
In general, the hk's are
convenient to define the dimensionless variable
x within each region k as
z hk
x
(1.9a)
so that region k can be described in terms of z as
zk < z < zk + hk = zk+
(1.9b)
or equivalently in terms of x as
0 < x < 1
(1.9c)
for each of the regions k, k-1, K.
This kind of dimensionless
variables will be used interchangeably with the original length
variables.
21
Now the Hermite interpolating polynomials, which are
polynomials of degree (2m-1), can be expressed in terms of a set
These basic functions are
uk (x) and uk (x).
of basic functions,
defined by the following rules:
1)
They are polynomials of degree (2m-1) in x.
2)
The highest value of p is (m-1)
3)
The u
(x) are zero except in the interval hk on the (+)
side of zk; similarly the uk~(x) are zero except in the
interval hk4)
on the
(-)
The p-th derivatives of u
side of zk'
(x)
are unity at zk and
zero at all other mesh interfaces.
5)
Derivatives of u
(x), lower than the p-th derivatives,
are zero at all mesh interfaces.
Using these rules, we can construct the basic functions for
Hermite interpolating polynomials of any odd degree.
Specifically,
for m-l (p=O only), we have linear Hermite basic functions:
u0-(
k
x,
k-l < z < zk
all other z
(1.10a)
0,
0+
uk W
1-x,
0,
zk < z < zk+l
all other 2
(1.10b)
and for m = 2 (p = 0 and 1), cubic Hermite basic functions:
0-
3x2 - 2x3 , zk-l < z < zk
() = 0
uk (x)
h(1.lla)z
al
all other
22
0+
.1-3x2
+ 2x3
zk
z < zk+l
(1.1b)
all other z
uk (x)=f 0
2
3
1+ -x2 + x3
ul+X
k (x)={0
1+
x - 2x
Uk (x){ 0
2
zkl< z < zk
k-1l
k
all other z
+x,
3
(1.llc)
zk < Z < zk+1
all other z
(1.lld)
The forms of these linear and cubic basis functions are illustrated
in Figure 1.1.
The Hermite interpolating polynomial of order m within a
region zk < z < zk+l is
just a linear combination of the basis
functions of the corresponding order:
m-1
H (x)
[a
=
(1.12)
u(x) + ai+luk+l(x)]
p=0
where a
's
and ai's
are constants which can be easily related to
the values and derivatives at (+) side of zk and (-)
respectively,
of whatever the function is
For two dimensional problems,
side of zk+1*
being approximated.
the Hermite basis functions
can be found by combining all the possible products from two sets
of univariate basis functions, each representing the basis functions
in
one direction.
23
Linear Hermite Basis Functions (m = 1)
p = 0
1
0uk
0+
(x)
Uk
zk-l
zk+1
Cubic Hermite Basis Functions (m = 2)
p = 0
0+
Z (x
0uk (X)
zk-1
zk
zk+1
p = 1
slope =1
27
-
r k (x)
zk-l
zk+1
4
1uk
Figure 1.1
~27i
Linear and Cubic Hermite Basis Functions
of Equations 1.10 and 1.11
24
1.4
Organization of the Thesis
The rest of the thesis is organized as follows.
Chapter 2
describes the use of a variational principle in time-independent
neutron diffusion theory.
The difference equations of some low
order finite element methods applied in one-dimension are derived
from this principle to illustrate the use of this technique.
Chapter 3 is devoted to some consequences of using simple finite
element trial functions as approximations and to an attempt to
solve the discontinuity problem at singular points.
The forms of
the finite element synthesis approximation in two-dimensions and
the derivation of difference equations are given in
with the results of some sample problems.
used are also discussed.
Chapter 4 along
The numerical techniques
Finally, Chapter 5 presents conclusions
derived from this study and the possibilities of extending the
method to more general cases.
25
CHAPTER 2
VARIATIONAL PRINCIPLES IN NEUTRON DIFFUSION PROBLEMS
Applications of the calculus of variation are concerned
chiefly with the determination of maxima and minima of certain
expressions involving unknown functions.
mechanics can be stated in
Many laws of physics and
terms of some kinds of minimization and
thus can be formulated in certain forms of variational principles
[33],
An important example is well known in classical
[37]-[40].
mechanics; Hamilton's (variational) principle,
t2 L
6
dt = 0
(2.1)
t1
is equivalent to Lagrange's equation of motion,
L
aq
d
0
dt9
(2.2)
0
where the Lagrangian L is a function of the generalized coordinates
q
and the velocities q
and time t.
Lagrange's equations are, in
turn, equivalent to Newton's.
Variational methods are particularly useful for determining
the approximate solutions of problems when the true solutions are
difficult to obtain.
Given the space of trial functions, the
variational methods will pick a "best" one from such a space
automatically.
In particular, if the true solution is within the
26
space of trial functions, the variational methods will find it as
the "best" solution.
Essentially, the variational method seeks to combine known
trial functions into approximate solutions through the use of a
characteristic variational functional.'" Thetef ore . the -f irst
step of the method is to find the characteristic functional whose
first-order variation, when set to zero, yields the describing
equations and their associated conditions of the system as its
Euler equations.
A space of trial functions, given in terms of
known functions and unknown coefficients (or functions), is then
chosen to approximate the solutions of the describing equations.
These approximate trial functions are then substituted into the
variational functional.
Setting the first variation of this
functional to zero and allowing arbitrary variations in all the
unknowns in the trial functions results in a set of simultaneous
equations among the unknowns.
Solving of this set of equations
yields the "best" obtainable approximate solutions within the space
of trial functions given.
Variational methods can be thought of as a kind of weighted
residual method since weighting functions appear in the functional
and in the equations that result from setting the first variation
of the functional to zero.
The weighting functions are determined
by the form of the variational functional itself.
In the case of
known adjoint equations, such as in diffusion theory, the
27
functional is usually chosen so that its Euler equations include
both the original describing equations and adjoint equations.
The
inclusion of corresponding adjoint trial functions in the functional
results in
adjoint weighting in
the variation equations and allows
greater approximation flexibility for the variational method.
2.1
Variational Principles in Diffusion Theory
The time-independent multigroup diffusion equations, given
by Equation 1.4, can be written as
(D
IID = 1(2.3a)
where
F=-
V - ID V + 1-
(2.3b)
T
The corresponding adjoint equations are
H*@* = 1
(2.4a)
**
where the adjoint operators, 1H* and F*
are defined as the transpose
of the corresponding operatorslH and IF, respectively:
1
= I T*= -
since U and I4 are diagonal.
(2.4b)
- rT
*-DV+3M
*
is
the group adjoint flux vector, or
importance vector, which must obey the same boundary conditions as
@[41].
28
The exact solutions 0(r) and 0*(r) of the diffusion equations
and the adjoint diffusion equations can be approximated by flux and
functions U(r) and U*(r)
adjoint flux trial
using a variational
functional of the form
=
JRU*U
u*T [ lilT
-
(2.5)
FU Idr
where it is assumed that the group flux trial function vectors U* and
U as well as the normal components of the group current vectors IDVU*
and
IVU across interfaces are continuous, and that U* and U vanish
at the outer surface of the reactor region R.
arbitrary trial function variations for the
making
F1
If we denote the
adjoint flux by 6U* and
stationary with respect to U*, we have
6F1 =R
u*T DHU
-
IFUIdr =0
(2.6a)
which contains the desired Equation 2.3a as its Euler equation.
Similarly, since Equation 2.5 can be written as
F1
making
F1
JfR U[T]T
I9FTU* ]dr
=R UT [*U* -
F*U*]dr
stationary with respect to U gives
UT[ fl*U* - -IF*U*]dr = 0
U=
29
(2.6b)
The
where 6U is an arbitrary variations of flux trial function U.
Euler equation of Equation 2.6b is evidently Equation 2.4a.
Thus,
we see that the exact solutions along with the exact eigenvalue are
reproduced if the exact solutions are contained in the given space
Generally, however, the trial function space
of trial functions.
does not contain these exact solutions and only approximate solutions
and eignevalues are obtained from invoking Equations 2.6.
Thus, the accuracy of the above approximation depends solely
Each
on the forms of the flux and the adjoint flux trial functions.
trial function can be defined in terms of known functions and
unknown coefficients (or functions).
Independent variations of
the unknown coefficients of the adjoint trial functions in Equation
2.6a will then yield the "best" flux solution for that class of flux
The corresponding "best" adjoint flux
function.
and adjoint trial
solution is found in a similar way by using Equation 2.6b.
The variational functional
F1
is not the only functional that
produces the desired variational equations 2.6 when made stationary.
Anohter functional incorporating the flux and adjoint flux diffusion
equations is the Rayleigh's principle [42],
RU* 3HUdr
F2 [U*'U]
1
xfRU'd
Although the forms of
functional
F2
(2.7)
-
U* 3FUdr
F2
and
F1
differ, it can be shown [43] that
gives identical Euler equations as Equations 2.6 and
is equivalent to functional
Fl.
However, because the form of
30
F1
is much simpler than that of
F
it will be used later in this
thesis.
2.2
Discontinuous Trial Functions
In the previous section we stated that the admissable trial
functions for flux and adjoint flux used in functional
F1
must
satisfy certain continuity conditions inside the reactor region R.
This restriction greatly reduces the flexibility and generality of
our methods.
In this section the class of allowable trial functions
will be extended to include discontinuous flux and current trial
functions.
Special provisions must be made in the approximation method
such that this extended class of trial functions can be properly
used.
In order to account for the discontinuities in the trial
functions, it is necessary to include special terms specifying
continuity conditions directly in the approximation method.
This
can be achieved through the use of a variational functional whose
Euler equations include not only the original P-1 equations but
also the associated continuity conditions for both flux and current.
A general functional of this type which allows discontinuous flux,
current and adjoint trial functions is given [18],
follows:
31
[44]-[46] as
F3
=
+ Aul + V*T. [VU +D_'V]1dr
l
{U*T
.U*,U,V*,v]
y 4 +U*)
+V*)
_V+T (U+ - U) ]ds
n -[ (U*
+
(2.8)
r
where U*, U, V*, and V are the group flux and group current approximations to D*, @, J*, and J, respectively, and where the first integral
extends over all r inside reactor region R and the second extends
i is the
over all interior surfaces where discontinuities occur.
unit vector normal to interior surfaces.
is pointing are denoted with the
sides of surfaces toward which
subscript (+)
Quantities evaluated on
and evaluated on the other sides of such surfaces are
denoted with the subscript (-).
The first variation of
quantities can be found in
3F*=
R
+
.
F3
with respect to the adjoint
a straightforward manner:
1
T[v.v + AU] + 6V*T. [VU +]D V]ldr
[ (6U* +
6 U*)T (-
V_)+(6y* + 6V*)T(U-U_) ]ds
(2.9)
r
The desired P-1 equations, Equations 1.6, and the associated continuity conditions for flux and current at the inner surfaces follow
directly by setting 6F
= 0.
To show that the adjoint P-1 equations
and the continuity conditions for adjoint flux and current also
resulted from this variational functional if
with respect to variation in
F3
is made stationary
the unstarred quantities,
32
it
is
necessary to rearrange Equation 2.8.
This can be done by integrating
by parts terms involving spatial derivatives and replacing all
terms by their transposes.
F
[U*,U,1*,V3
=
The result is the alternate expression
{UT[-.VV* + A*U*1+
lf..T
-
+]U)-V*]}dr
-[-_.V*
- [(U+ + U_)T
-+ + V..)
++ - _V*)+(V
f (V
T
It is clear by now that if the first variation of
(U*
+ - U*)]ds-(2.10)
(.0
F3
with respect to
its unstarred arguments is now set equal to zero, we get adjoint P-1
equations along with their continuity conditions:
- V-J* + A*$* = 0
(2.lla)
VO* - D~1J* = 0
(2.11b)
*=
.
(2.110)
(2.lld)
.J*
=
In most applications, only approximations to the flux and
In such cases variational functional
current solutions are desired.
F3
in the form of Equation 2.8 is more convenient and the approxima-
tion is
then based on the following:
6uT [V+
fR
1T
+
r
AU] + 6V*T-LU + ID l] )dr +
- [ (SU* + 6U*)
(
- V-)+(6V* +6V*)
T
(U+ - U_)]ds=0 )
(2
+
(2.12)
r
33
If the adjoint trial functions are defined as
U* = U
(2. 13a)
V* = V
(2.13b)
then Equation 2.12 reduces to the Rayleigh-Ritz Galerkin method
[47]-[49],
a weighted residiml method based upon flux weighting.
The variation equation can be further simplified for certain
cases regardless of the choice of weighting.
For the approximation
methods which require the currents to obey explicitly Fick's laws:
-VUDU
(2.14a)
V*= + DVU*
(2.14b)
V
Equation 2.12 reduces to
R[6U*AU
-
fi.[(6U*
+
]DV]dr_
6V*T
-
6U*)T (V + V_)+(6y + 6_V)
(U+- U_)]ds = 0
r
(2.15)
If in addition the flux and adjoint flux are required to be everywhere
continuous,
the above equation reduces to the simple form
R[6U* AU -
6 V*T
,
-
dr
=
0
(2.16a)
or equivalently,
f [6U** AU + (&6 U*)- ID(VU)]dr = 0
34
(2.16b)
One note about the boundary conditions:
In the present
discussion the allowed trial functions used in the approximation
methods must satisfy the same boundary conditions as the true
solutions.
Though it is possible to extend the variational principle
so that the permissible class of trial functions is augmented to
include functions with arbitrary conditions on the boundary, we
shall not pursue this point since the imposing of boundary conditions
is
easy with the use of Hermite interpolating functions as trial
functions.
2.3
Fick's Law as a Consequence of Variational Principle
Some simple derivations of the difference equations are given
here using trial
dimension.
functions of finite element methods in
one-
These derivations will show that Fick's law always sur-
faces as a natural consequence of the variational equation 2.12. In
each case, flux and adjoint flux continuity is assumed in the
approximate trial functions while current continuity is not.
Various
forms of current trial functions are used in different cases.
In
the derivation process we shall first allow the values of trial
functions at boundaries to vary arbitrarily, then impose the required boundary conditions (which usually involves setting some
boundary values to zero and eliminating some equations) after we get
the general system of equations.
This procedure is totally
equivalent to that of first imposing boundary conditions and then
35
finding the desired system of algebraic equations, and will be used
throughout this thesis.
In this way we can accomodate various
sets of boundary conditions at one time.
2.3.1
The Linear Basis Function Approximation
The group flux trial functions defined as nonzero within each
region k can be expressed in terms of linear basis functions as
Uk (z) = (1-x)Pk +
k+
(2.17a)
k = 1 to K
U*(z)
(2.17b)
= (1-x)Pk + xPl
and P* are the approximate group flux and adjoint flux
where P
k
k
column vectors, respectively, at point zk, and 0 < x < 1 within
each region k.
Figure 2.1 illustrates the form of these trial
Different forms of current trial
functions.
Fick's law current,
We shall start
with (i)
current,
linear current, and (iv)
(iii)
functions can be used.
followed by (ii)
constant
quadratic current trial
functions within each region k.
(i) Fick's Law Current Trial Functions:
The group current trial
functions which obey Fick's law are
given, according to Equations 2.14, by
Vk(z) = hIk(x)[Pk
2.18a)
k+1]
; k =1 to K
V*(z)
k
=
hz
k
Ekj()[P*
P*]
k
(2.18b)
36
zk-1
Figure 2.1
zk
zk+l
The Flux Trial Functions of Linear Finite
Element Approximation
37
Insertion of these trial functions into variation equation
2.16a results in the equation
K
k=1
1
k
{[(l-x)P
0
+ [6P
- 6Pk+1
+
k(x)[(x)pk + xP
xk+
1
- P
]
1J}dx =0
(2.19)
hk
Allowing arbitrary variations in all P* results in a system of
k
K+1 equations in K+l unknowns which can be written as:
b P
+ dP2
k + Ck k+l
akkl +k
=
0
(2.20a)
0 ; k = 2 to K
(2.20b)
(2.20c)
aK+1PK + bK+ PK+l =0
where the GxG matrix coefficients ak, bk, ck are of the form
A
-
B and are defined in Section 1 of Appendix C assuming
homogeneous regional nuclear constants.
Zero flux boundary conditions
can be imposed by use of only Equation 2.20b with P1 = PK+1 = 0'
while symmetry boundary conditions require the use of the other
equations as well.
The matrix form of these equations for the
boundary conditions of zero flux on the left and symmetry on the
right is given in Figure 2.2.
(ii)
Constant Current Trial Functions:
The current trial functions which are constant within each
region k are given by
38
2
2
1
k
6k
Sk
+1
K+1
5k
k
K+1
PK+1
K+l
L~)
~0
Figure 2.2
Matrix Form of the Linear Finite Element Method Approximation.
Eqs. 2.20 for the case of zero flux on the left and symmetry
boundary condition on the right.
Where:
ak
b
ck
=
k
X
k
X
1
k
k
1C
k =2 to K+l1
PK+l
(2. 21a)
Vk(z) = Qk
k = 1 to K
;
(2.21b)
= Q
Vk(z)
where Q and Q* are column vectors with constant elements representing
the approximate values of group current and adjoint current,
Since Fick's law is not true in
respectively, in the region k.
this case, Equation 2.12 must be used instead of Equation 2.16a.
Insertion of Equations 2.17 and 2.21 into variation equation 2.12
yields the equation
1-x)6P
f {[
0
h
k=l
+
6Q*T [-(Pk
hk k
k
+1
+ x6P*
k+
kA(x)[(l-x)P
+
]T
kxkQk]d
~
K
+ k 2k-1)
=
0
(2.22)
Allowing arbitrary variations in all Q* gives
f[Pk+
.*.
k) + hk Dk
[j
xQkdx = 0;
(x)dx]Qk ~
k
k+l
k= 1 to K
k
or equivalently,
k
Q
Vk(z) =
1:klxdx(Pk+l k)
1
(x)
x]
-[IfIDk(x)dx]
-d
1
-
[
dzk
0
40
U(z) ; k
=
1 to K
(2.23)
Equation 2.23 can be thought of as Fick's law in an integrated sense.
In the case of constant diffusion coefficients within each region k,
Equation 2.23 reduces to the simple form of Fick's law:
Vk (z)=-]Dk d
Uk(Z)
(2.24)
in each region k.
Allowing arbitrary variations in all P* results in a system
k
of equations of the same form as Equations 2.20,
PkIs for Qk's by using
different in general.
Equation 2.23.
if
we substitute
The matrix coefficients are
However, for the case of homogeneous regional
nuclear constants, they are identical to those given in Section 1
of Appencix C.
(iii)
Linear Current Trial Functions:
The linear current trial
functions which are not necessarily
continuous across the inner interfaces between regions can be expressed in
terms of linear basis functions as
(2.25a)
Vk(z) = (1-x)Qk,+ + xQk+l:
k = 1 to K
+ xQ*
V*(z) = (1-x)Q*
10
k9+
k
(2.25b)
where Qk,+ and Qk+1,- are the approximate group current column vectors
at positive side of node zk and negative side of node zk+1, respectively.
The form of these trial
functions is
illustrated in
Figure 2.3.
Substitution of Equations 2.17 and 2.25 into variation equation
41
2.12 results in
the equation
T
] {
K
<[(1-x) 6P* + x6P
hk
k=1
(k+1,
h
k
0
+ A(x) [ (-x)Pk
K
1r
k+
hk <[
kf(1-x)6Q*
+
0
k=l
1
x6Qk+,
+ X~k+19-
+ 1D-
+k
K
+ Xpk+l}>dx
1
k+
k+l
h
pk
lI>dx
WxI[(1-x)Qk
k+1l,-]}d
k,+++ xQk
T
+
6 PT
(
k
-
(2.26)
=
k=2
Allowing arbitrary variations in all
k) + [J(1-x)
21(Pk+1
+[J
(1-x)xIDk
2
Q
1(Pk1
+l
P)
+[1
x2
[J
+
1
we have
1j1 (x)dx]Qk,+
(x)dx]Qk+,
Similarly, arbitrary variations in all
2hk
,
k = 1 to K
- 0;
Qk1,
(2.27a)
gives
x(l-x) ]D~'(x)dx]Qk,+
(x)dx]Qk+1,- = 0;
42
k = 1 to K
(2. 27b)
Solving simultaneous Equations 2.27 results in
Qk,+
ks+
(1-x)x ek (x)dx]
1jk
[ix]D(x)dx]-[i
k 0(k+lk)
2h k k+
k
0
}(P1
0
(2.28a)
(x)dx][fl~-x)'D~l~~dx]'-[f(l-x)x]Dk
[
x)dx]
Q IBk+12 hk
Qk+1,-
k+1,-
where]Bk,+ and
-
are GxG diagonal matrices defined by
]Dk1 (x)dx]
-l(x)dx]l 'x(1-x)
Bk,+ rfx2
0
(2.28b)
k)
(Pk+1
0
[fx(l-x) Ek
0
1
1[f-(x)2 D- 1 (x)dx]
(x)dx] l
-1lx)dx] 'fl~
EBk+1,- =[lx2
-[f x(l-x) IEkl(x)dx]l
0
If E)k(x) is
(2.29a)
0
fx2
0
-x]Ekl x)dx]
-l(x)dx]
(2.29b)
constant within each region k, then Equations 2.29
and 2.28 can be reduced to
B
-1 -1 1
B1
B k+1,-
=Bk,+
Qk+1,-
k,+
IDk
-1
IDk
1 -1 -1 1
-1
kIDk
3
V
and
(Dk
so that the current trial
1-
1
k+1
k
k
k+1
functions become constant and are given by
43
Vk(z)
=
,
-
Uk Uk(z); k = 1 to K
Ik (Pk+1-
h
(2.30)
i.e., the current trial functions are related to flux trial functions
via Fick's law.
The system of equations relating all Pk's can be found by
allowing arbitrary variations in all Pg's in Equation 2.26 and sub-
k
stituting for all Qk,+'s and Qk+1,-'s with Pk's.
Under the circum-
stance that all regional nuclear constants are homogeneous, Equation
2.20 with matrix coefficients defined in Appendix C.1 results.
One interesting situation arises when some regions are
symmetric.
For example,
k(1-x) for 0 < x < 1.
Qk+l,-
if
region k is
symmetric,
then Dk(x)
Substract Equation 2.27b from Eq. 2.27a gives
Qk,+ since
f1lx)2 U1 (x)dx
0
(1-x) 2 Dk 1 (1-x)dx
=
=
x2 D-1(x)dx
0okf
0
by simple change of variables.
1
Therefore,
the current trial
functions
become a constant in this region and is given by
Vk (z)=Qk,+
-
([x]D
x)dx] ~(k
d
f1 x
kl(x)dx] 1
(Z
SUk(z)
which, as it must, reduces to Fick's law if Dk(x) is constant.
44
Quadratic Current Trial Functions:
(iv)
The current trial functions which are quadratic in general are
given by
Vk(Z) = x2 Lk + xMk + Nk
(2.31a)
k= ltoK
+ xMI + N
V*(z) = x2L
where Lk, Mk,
(2.31b)
Nk and their corresponding adjoint quantities are all
Gxl column vectors with constant elements.
They are not continuous
across the region interfaces in general and are illustrated in
Figure 2.4.
Insertion of
Equations 2.17 and 2.31 into variation equation
2.12 results in the equation
1
K
k,=1
hk
0k
T
<[ (1-x)
6
P
+ x6k+1
{(2xLk+Mk
k(x)[(l-x)Pk
Pk+l ]d
k++ xk+1]}>dx
Ak
++A
K
+ Ih
k=l
1
2Tl
<x26L + x6M + 6N*]Tk(P k+lPk
2
+Dk 1l
+
K
I 6P* [N
-
(L
2Lk + xMk + Nk]}>dx
+ Mk1 + Nk-1)] = 0
(2.32)
k=2
Allwwing arbitrary variations in L , Mg,
equations
45
and N* yields a set of
zk-1
Figure 2.3
zk
zk+l
The Linear Current Trial Functions (as defined
by Equation 2.25a)
zk-1
Figure 2.4
zk
zk+l
The Quadratic Current Trial Functions (as defined
by Equation 2.31a)
46
2{
hkf
0
k+1 k) +
2
+ xMk + Nk])dx - 0
(2.33a)
x 11 (Pk+l k) +]Dkl (x)2Lk + xMk + Nk])dx = 0
hk
(2.33b)
hk
k+1
k+l- k
hk
hkfl ~h
0
()[
L
N
+~k
- (x)[x2
(x[ Lk + xMk + Nkd
(2.33c)
=-]dx
0
k
k = 1 to K
which can be solved for Lk, Mk,
the case of constant
and Nk in terms of Pk and Pk+1.
For
k in each region k, Equations 2.33 have
the solution
k
Nk
; k - 1 to K
=
kk
k+l
k
so that the current trial functions again become constant and obey
Fick's law.
Again, the system of equations for all Pk's is given by
Equations 2.20 under the assumption that all nuclear constants
are homogeneous within each region.
2.3.2 The Cubic Hermite Basis Function Approximation [50]-[51]
The group flux trial functions within each region k can be
expressed as a linear combination of cubic Hermite basis functions
as
47
Uk(
= (1-3x 2+2x3
k + (3x 2 -2x3 )Pk+l
2 3
-123
-(x-2x +x )h D (x)Qk
U*(z)
2 3
(1-3x +2x )P
=
-(x-2x2 +x3 )hklD
where k = 1 to K.
P
k
x
2
+ (3x -2x
W Qt
3
)h
D(x)Qk+
(2.34a)
+1
-
2+x3 )hk]D-1 (X)Q+
(2.34b)
is the approximate group flux solution vector at
node zk and the meaning of Qk will become apparent later.
Because
of the properties of basis functions, continuity of flux is automatically met since
Uk(zk)
=
Uk(x=O)=PkUk-1(x)=Uk-l(zk)
U*(zk)
=
U*(x=O)=P*=Uk(X=1)mU-l(zk
Figure 2.5 illustrates the forms of these trial functions.
For the current trial functions we shall again assume various
forms starting with (i) Fick's law current and followed by (ii)
Quadratic current and (iii) Linear current approximations in each
region k.
(i)
Fick's Law Current Trial Functions:
Application of Fick's law defines the current trial functions
for each k as
48
zk-1
Figure 2.5
zk
zk+1
The Cubic Hermite Basis Functions
Approximation (as defined by Equation 2.34a)
49
Vk(z) = hDk(x)( 6 x- 6 x 2
k~k+1)+(1-4 x+3 x2 )Qk+(-2x+ 3x 2
k+1
(2.35a)
+(P*
1 -P*)+(-l+4x-3x 2)Qk+(2x-3x2 )Q+ 1
Ek (x)(6x-6x2
V*(z) =
(2.35b)
with x inside each
where the assumption that Dk(x) varies very little
region k has been made.
From Equations 2.35 it is clear that Qk is
the approximate group current solution vector at node zk and that
current is continuous across the region interfaces:
Vk(zk)
=
Vk(xO) = Qk= Vkl (xl) = Vk-l(zk)
V*(zk
=
V*(xinO)
V (z
=
kk
-=
V*
k-1
(X-l)
=
V*l(zk
k-l zk
Equations 2.34 and 2.35 are now substituted into variation
equation 2.16a and the resultant lengthy equation can be written
as follows:
6P* (bl P
+ b2 Q + dl P
6Q*T {b3 P
+ b4
1
+
+
b 1Q1
Q
l1P2
+ C2
lQ
+ c3 P2 + c41Q2
T
K
+
1 1
6 P{al
K
I6Q* {a3k k-
k=2k
+akk-
+ a4k k-
blk
+
+ clkPk+1 +
kk+1
+ b3*Pk + b4k k + c3k k+
kQk+l
k k+l + c4k~+
kk
k
50
+
P
{alK+1 K + a2K+lQK + bK+1PK+l + b2K+lQK+l}
1
+ 6Q*
{a3K+
K + a4K+1QK + b 3K+lPK+l + b4K+1QK+l
=
(2.36)
where GxG matrix coefficients {al,---,c4} are of the form A -
and
are defined in Section 2 of. Appendix C assuming homogeneous regional
nuclear constants.
Allowing arbitrary variations in all P* and Q* results in a
equations in
system of 2(K+l)
zero flux, Pk
k
- 0, or zero current,
P
0 as well as
=
~
kk
SQ* = 0, boundary conditions for k
well as
The choice of either
2(K+l) unknowns.
k
Qk
=
0 as
1 or k = K+1 reduces the
-
Figure 2.6 illustrates
system to a set of 2K equations in 2K unknowns.
the matrix form of such a system for the case of zero flux on the
left and zero current on the right boundary conditions.
(ii)
Quadratic Current Trial Functions:
As in Section 2.3.1(iv), the current trial functions which are
quadratic are given by, Equations 2.31.
The insertion of Equations 2.34
and 2.31 into variation equation 2.12 yields
K
kl2
h
k=l
3
*+
<[(1-3x +2x
3 )hk 3D~(x)Q+
2
+( 3 x2
36p
2
1
D~lx)hk*]
23
)k+1
-
+ (-x+2x -x )hk ]D
k]l
kj
0
+(x2_
2
+ (3x -2x
] {
(2xLk4
h
(x)6Q
k)+x) [(1-3x2 +2x3 )k
k
x )pk+l + (-x+2x2 x 3
dx
51
k
(x)k(x
k+(
2
(x)
_3
1
K+l
Q1
c4
b4
c3
a22
bl
2
b2 2
c12
a42
b32
b42
c32
c2
2
Q2
c42
-I
L
alk
a2k
blk
b2k
clk
c2 k
a3k
a4k
b3k
b4k
c3k
c4k
= 0
U'
alK
a2 K
blK
b2K
ClK
a3K
a4K
b3K
b4K
c3 K
alK+1 a2K+1 blK+l
Figure 2.6
Pk
P K+l
Matrix Form of the Cubic Hermite Finite Element Method Approximation
Equation 2.36 for the case of zero flux on the left and symmetry
boundary conditions on the right
where
ank = ank ~
bnk =
cnk
=1
nk '
k
1nk
En
~kX
nk
1 to 4
and
k= 1 to K + 1
n
=
K
+
1k
l < x 6
2
+
~
x
, ] {
+
{
_
2
[(6x-6x2
2k-12
k+l~k)
-
+ (-l+4x-3x2)hkDk (x)Qk + (2x-3x2
1hk
(x)Qk+l
Skl(x)[x2Lk + xMk + Nk}>dx
+
K
I 6P*
k=2
[Nk-(Lk-l + 1-kl + Nk-1] = 0
(2.37)
Assuming thatID k(x) is constant within each region k and allowing
and N
q
arbitrary variations in all L , M
results in the following
equations:
k+
h kEt
kl
k. +D
(
Lk +
2
)
1
12
)
+ 1
k-lk+
1
-3kmkQkl0kk
11
k+1
+ ID 1 (1 L
1
h
Mk +
Nk)=0-
-1
kk
M
+
k+l]
1
122k
hkk.
+
k+l1
Nk
1
-1 1
-P'
(
Mk +
(Pk+l k) +Dk ( Lk +
0
)
which can be readily solved to give
Lk
hWkk
Nk
-P M
h k(.8k
Nk
k+1-)
+
pso
k1
k + Qk+1
2(2Q
k+l)kk+l
+
)
(2.38)
=k
53
functions in
so that the group current trial
Vk
1
this case becomes
k(6x-6 x2 )(Pkk+1)+(l-4x+3xQ+(-2x+3x2 )k+l
i.e., the Fick's law current.
It can be shown that Equation 2.36 with identical matrix
coefficients, as defined in Appendix C.2, is the result of taking
arbitrary variations in all P* and
k
in Equation 2.37 and subQ*
k
stituting for Lk, Mk and Nk using Equations 2.38.
(iii)
Linear Current Trial Functions:
To make the notation more&clear, the linear current trial functions
given in Equations 2.25 may be rewritten in the form
Vk(z)
(2.39a)
= xMk + Nk
; k = 1 to K
V*(z)
=
k z)xTj
(2.39b)
+ N*
k
where Nk is the approximate group current column vector at positive
side of node zk and (Mk + Nk) is the approximate group current vector
at negative side of node zk+l*
Substitution of Equations 2.39 and 2.34 into variation equation
2.12 results in the equation
54
<[ (1-3x2+2x3)6P*+(3x2-2x3)6Pk+ +(-x+2x 2 _
k
k=-
+(x2
3 )hklk
6Q*ITk1
k+1]
(
+(-x+2x2_ 3 )hk
K
+
hk
-1
0
l
2
0
k=1
k+l
k + (x2-x )hk Dk (x)Qk+l ]}>dx
T
<[x6M:+ 6N ]T{
2 hk
(x)[ (1-3x2+2x3)k+( 3 x 2-2x3
2
k)+(-l+ 4x-3x2
)k+1
[ 6x-6x
-l
k
k
-1
-1
2
+ (2x-3x,)hktDk (x)Qk+I] +]'Dk (xX 'k+Nk)}>dx
+
K
I SP
k=2
(2.40)
- k-l + Nk-1)] = 0
[N
Assuming that IDk(x) is constant within each region k and taking
and N* yields the following equations:
arbitrary variations in
1
k) + L- hk
[i(Pk+1
1
+
hkPk+lk) + ]k
k
-P-P
11
k1]
+k
+
=0
+Nk)
(Mk
(
Mk +
which have the solution
=
N
k
- k
Qk+l
-P
(P
k k+l k) + 2(kk+L
-0D
hk
The current trial
Vk(z)
=
k
k
k
functions then take the form of
k+
+k (
- t)
55
Qk+l
~ Qk)
Nk)
00
which is not Fick's law.
This result is expected since the degree of
the assumed current trial functions is not high enough.
2.3.3
Summary of Section 2.3
From the study of the previous two subsections, we see that the
use of variational equation 2.12, which is derived from setting the
first variation with respect to adjoint quantities in the variational
functional
F3
to zero, has the ability to force the current trial
functions to obey Fick's law in one-dimension provided the following
conditions are imposed:
1)
The flux (and adjoint flux) trial functions are continuous
across the region interfaces,
2)
The diffusion coefficient matrix ]Dk is constant within
each region k, and
3)
The degree of current trial functions is higher than or
equal to the degree of flux trial functions minus 1.
It can be shown (through more lengthy algebra) that the above conditions also apply to two-dimensional problems.
In this case, however,
we must change the third condition to say that the degree of X-direction
current trial functions in x must be higher than or equal to the degree
of flux trial
functions in: x minus 1 and the degree of flux trial
functions in the y must be higher than or equal to the degree of flux
trial functions in y.
An analogous condition must be satisfied by
Y-direction current trial functions.
56
In the next two chapters, we shall assume that the approximate
current (and adjoint current) trial functions are always related to
the approximate flux (and adjoint flux) trial functions via Fick's
law, whether the latter are continuous acorss the region interfaces
or not.
However, when it is possible to make the flux trial functions
continuous throughout the reactor (as in Chapter 3), we shall assume
that
too, so that the simpler variation equation, Equation 2.16a,
can be used instead of Equation 2.12 in the derivation of the
difference equations.
57
CHAPTER 3
INVESTIGATION OF FINITE ELEMENT METHOD USING DISCONTINUOUS
CURRENT TRIAL FUNCTIONS
For a one-dimensional problem, the use of linear Hermite interpolation polynomials as approximate flux trial functions in the
form of Equations 2.17 limits the number of unknowns to one for each
node.
Since these unknowns are tied up with the approximate values
of the flux at nodal points z = zk,
imposing the current
this leaves us with no choice for
continuity conditions if Fick's law current
trial functions are used.
The same situation also exists in two-
dimensional problems if we use bi-linear Hermite interpolation
polynomials as flux trial functions and demand that they be continuous
across all region interfaces.
The approximation employing cubic Hermite basis functions in
one-dimension, as described in Section 3.2 of Chapter 2, has the
ability to match both flux and current at interfaces between adjoining
regions if
Fick's law is
assumed.
Extension to using bi-cubic Hermite
polynomials as flux approximations in two-dimensions, however, encounters the problem of current discontinuity in that no matter what
forms of approximate trial
functions are used for the flux, there is
no way to force the Fick's law current trial functions to be continuous at those points which are formed by intersections of two or
more material interfaces [52].
These points are called singular points.
58
In order to make the approximate solution to the diffusion
problem more complete,
solutions
it
[53]-[54].
[47],
would be necessary to include the singular
However, this is an impractical task for
computation and the approach to be taken in this thesis is to ignore
the singular part of the solution.
The effect of singular solution
on reaction rates and integral properties in reactor problems is
generally negligible.
It is the aim of this chapter to investigate the consequences
of not imposing any condition on those "current" coefficients in the
cubic (or bi-cubic) flux trial functions, thus completely relaxing
Use of the variational
the requirement of current continuity.
functional
F,
which permits the discontinuous trial functions in
its solution space, whould enable us to find the "best" relation
between these floating parameters.
For simplicity, and without loss
of generality, we shall assume that the nuclear constants are
homogeneous within each individual subregions.
3.1
Derivation of Difference Equations in 1-D
The approximate cubic flux trial and weight functions, which
are continuous throughout the 1-D problem domain, can be expressed
in the following forms:
Uk(z) = u
U*(z)
k
=
=
(x)Pk + u0(x)Pk+1 + ul+ (x)k,+ + u1 (x)Qk+1,-
00+
u0 (x)P* + u0-(x)P*
1+
k+l + ul
k = 1 to K
59
Q*
k,+
1
+ u~(x)Q*
k+ ,1
(3.la)
(3.lb)
where the Uk
k,- and corresponding adjoint quantities
Sk,+'
*k'
are G-element vectors,
0+
and a ~(x) and u
1±
(x)
are cubic Hermite basis
With the understanding that they
functions defined by Equations 1.11.
are defined only over subregion k for each k = 1 to K, we can suppress
The current trial
the subscripts associated with them.
functions
defined by Fick's law in each region are given by
1
VZ
k(z)
-
du 0 -
du0
k dx
k +
k+1
Q+
k,+
dul1
dx
du1 dx k+,(3.2a)
P*
k+1 +d dx
dx
k + d0Pk
1
E
1\k1dx
V*(z)
k
Q*
k,+ +
dx
d
+1,
kl(3.2b)
k = 1 to K
Insertion of these trial function forms into variation equation
2.16a results in the equation
h
1
{[6Pu
klJO
k1 k0
ku
(x) + 6Pku
±+(xk+l
1+(x) +
k +u
+*1,-
()+uk+..u()
0+ (xPk + u0~(x)F k+1 + ul1+ ()k,+ + u 1~(x)Qk+l 1"k[u
k=l
*
1 f, du+ +
1 Q6
+ dx
k
k 0
du0~
du+
1) du0+P +du0dx
k
k dx
P
k+1
duk
dul
x
dul.+ +SQ* 1.du-T
+ 6Q*
k+l,- dx
k+
,1x- dx
du+1
1d
d - Q
u1+ Q1
Qk+l,-]}dx - 0
dx Qk,+ +
(3.3)
60
T
Allowing arbitrary variations in all P*,
k
=
1 to K; and Qk,-,
results in
1;
These equations are given
k = 2 to K + 1).
in Section 1 of Appendix D.
The GxG matrix coefficients in these
1
equations are of the form A
, and Q*k+1
3K+l unknowns (Pk, k = 1 to K +
a system of 3K+1 equations in
Qk,+,
Q
-
B where A includes diffusion,
absorption and scattering of neutrons and B includes fission neutron
production of certain regions.
The choice of either zero flux, Pk= 0 as well as 6P*=O for k=1 cr
k
K+l, or zero current, Q+=6Q*=
0 or
K+,-
= 0, boundary
=6Q
conditions reduces the system to 3K-1 equations in 3K-1 unknowns.
we arrange the unknowns in order of
Qk
If
k, Qk,+, then the resultant
coefficients matrix of this system of equations is a symmetric, 7stripe matri4.
3.2
Derivation of Differente Equations in
2-D
For two-dimensional rectangular geometry problem domain shown in
Figure 3.1, we subdivide the continuous variable X in
the horizontal
direction into I adjoining intervals and variable Y in the vertical
direction into J adjoining intervals so that each region (i,j) is
bounded by the lines X=X1 , X=X+1' YYj, and Y=Yj+ 1 .*
one-dimensional case, we define h
widths of region (i,j) in
=X i+1-X
and h
Similar to the
=Y j+y
X and Y directions, respectively.
as the
We also
define the dimensionless variables x and y within each region (i,j) as
61
(XI+1Y1)
(I,1)
(I,J)
(X1 p'J+j)
IX
+1'
hxi
h
Xi+1 - x
= Y j+1
yj
X - X
x
=
<X
i
hxi
i+1
Y - Y
y =
Figure 3.1
hA
Y
fY<Yj+1
in each of
the region
ij
yj
The Subdivisions of Rectantular Problem Domain
62
J+1
x
-X
(3.4a)
x
xi
Y -Y
h
y
so that region (i,j)
O
(3.4b)
yj
can be 'escribed in
x4l;
O1y
terms of x and y as
l
for each region (i,j), i = 1 to I, j
=
1 to J.
Because there are four cubic Hermite basis functions in each
direction, 16 bi-cubic basis functions can be formed for a twodimensional problem within a region.
The most general form of flux
trial function in any region (i,j) is thus a linear combination of
these sixteen basis functions.
Careful examination to satisfy the
requirement of flux continuity reduces the general form of flux trial
function in region (i,j) to
U
(x
3y)=[P u
(0,0)
+P ij+1
(xWu 0+(y)+P (0)u
0+
u
0-
(x) u
0 (x)u0+ ()+P,
(1,0)
(y).+P i;-(
1+
.+)U
0+
xWu
+1-(x U00
(1,0)
(y)+) +1,j j(-)U
+ +i+lsj+l
P(
) (-)u1 ~(x)u0(y) + P (10)(+)ul+
Wu0-(y)
itj+l
+ P (1)(+)u
+ P(u~
(x
Wu1+ (y)+P (1)(+)u0~(xWul1+()
u~yi+lo
(
x
(-)u 0 (Wu 1 -(y) + P(O l)(.H.uG+ (X)ul1-(y)
i+ltj+l(
i~j+l
+ P (0,1)
63
1(x)u
o+
(y)
(1)(l)u1+ (x)u1+(y)+P
+ P
+ P
(3)u1~(x)u
(2)u1 (- u 1+(y)
i
(y) + P( 1
1
i~l~j~ligj+l
u1+ (X)u1-(y)]
)
(3.5)
where P (00) is the approximate value of flux column vector at node
inj
(X1 ,X ),P
(+)and P (9l
f )*
vector at node (Xi,
Y ) and Y-direction column vector at node (XiYj+),
respectively, and P (l
values of
(k-1 to 4) are related to the approximate
XY at (x,
respectively,
(+) are related to X-direction current column
YJ), (Xi_,
Y
),
(Xi_, Y _),
and (Xi+, Y _),
Similar expression for adjoint flux trial
region (i,j), U*
functions in
(x,y) is formed by replacing all P's with their
19j
corresponding adjoint quantities (P*'s).
The current trial function in region (i,j) can be written as
(3.6)
(xy) = I Qi, (x,y) + J R 9(x,y)
(x,y) is the approximate X-direction current trial function
where Q
and R
(x,y) is the approximate Y-direction current trial function,
both in
region (i,j).
directions, respectively.
Q
(x,y)
ihjxh
iiaU
I--D
3x
ID
the X-
i and iJ are unit vectors in
i,j
According to Fick's law,
(x,y)
P(00) d
la~ ij
WX
0+
+
1(0)
du
i+l $j dx
U.(Y
(0,0) du 0 + 0du0- 0+ P(oO)
u (y) + Pi,j+l dx u (y)
i+l,j+1
dx
64
(y)
and Y-
+
P('51)
du
u (y)
dx
i9j
du1
P (1,0)
+i,j+1
+
19,j
+
duG
dx
+ p (01)
i+1,j+1
1+
du~
dx
+
du
(,)
dx
du0
(0 1)
(Y
+ (y) + P(1)
u
du
dx
(y)
(Y
dxU 1+
+(+)l ,
(0,1) (
i,j+1
1+
0-(
dul+
ij+1
(y) +
1-
+ 1 P (1) dul
(131)
P(0 () du0+
i+1,jx
0u
x
+
+
1-(
u
(2) dul
+(
1-(1)
P
( 3 )d
i+1,j+1
dx -u
(1,)
i"
g '±j+.(4) ddx
u
(y)]
~I(.a
and
R (x,y)
= -
-U U x (xy)y)
h
yj
_
-
j
(0,0)
u (x)
h,
i+j+
SP(1,0
:Loj
(x)
u1+
(+)
(+uC)dy
(1,9,0
i+1,j+1
dy
du0 ~ + P(Oo)
0-
(0,0)
+ p(0,0)
+i+,j
du
0+
1
dy
du~
dy
P
x)
(-)
(-
+
du0 +
dy
du0 ~
0+
u
du 0+
0-
dy
u1- W
Cx
du+
dy
1+ du0 ~
(
i,j+1
65
dy
(3.7a)
+ p
i+l ,j
dy
iti
du ~
0-
(0,1)
+ p1
dy
(-)
(1)
+
(1
+ P
1(3)
i+1,j +1
i~jdy
dy
du1~
du+
1-
dy
() u(x
(4) u1+
x
4)j~
u
+ P(
du
dy
(2)
1+
d
0+
(0,1)
+ Pij+
+ P
1
u,1)
(+) u 0~W
+ P (0')
u 0+(x) d
l(+)
3.7b)
du
d(.b
Similar expressions can be written for adjoint current trial functions
in
each region (i,j).
The variation equation 2.16a can now be written as
I
I
J
Jh
=1=1
h
1
Jdx
0
0
dy [6U*
T
A
U
(6 Q*T I-1
+ R*T ID - R
)]
i9j ±ji19j
ijj i9j i9j
-
= 0
(3.8)
Substitution of Equations 3.5 and 3.7 into this equation, then allowing
arbitrary variations in all starred adjoint quantities results in
a
system of equations involving (3I+1)(3J+l) equations in same number
of unknowns.
These equations are given in Section 2 of Appendix D.
The GxG matrix coefficients in these equations are of the form
A
-B.
to J + 1,
The ordering of unknowns follows the rule that: (i) j=1
(ii) for each j,
i=1 to I+1, and (iii)
for each (i,j), the
nine unknowns (less than nine at boundary points) are in
p(0'0), p(10) +9(1,0)_
1,3
i~j
i,3
. (0,1)
1,3
66
.,(0,1)
i~j
9(1,1)(i)
i~j
the order
P"(2'1)
i,j
, P2( ') (3),
i,j
and P (.)(4).
i,j
Imposing the boundary conditions reduces slightly the number of
equations as well as number of unknowns.
types of boundary conditions
However, regardless of the
imposed, the system of equations is
an NxN matrix problem of the form
iA P + 1
(3.9)
P_
A and B are independent of X and P is a column vector consisted
where
of N unknown elements.
The order N of the matrix equation is dependent
on the chosen boundary conditions and is given for various choices in
Table 3.1.
3.3
Numerical Method
The matrix equations which results from the approximations given
in this chapter are of the form of Equation 3.9.
The source iterative
scheme and the Cholesky method which are used in solving these equations
are discussed in this section.
In this Chapter, Equation 3.9 has been defined as ordered first
by spatial indexing followed by group indexing within each spatial
index.
For convenience, it is a general practice to reorder these
equations so that they are ordered first by group indexing followed
by spatial indexing within each group.
After reordering, Equation
3.9 can be written as
(]L +]M) P = TP +
B P
67
(3.10a)
Table 3.1
Matrix Order N of the Bi-cubic Hermite Basis
Functions Approximations with Discontinuous
Current Trial Functions
1 - Zero flux boundary condition
2 - Symmetry (zero current) boundary condition
Matrix Order
Boundary Condition Type
Left
Right
Top
Bottom
1
1
1
1
1
1
1
2
1
1
2
1
1
2
1
1
2
1
1
1
1
1
2
2
2
2
1
1
1
2
2
2
2
1
2
2
2
2
1
2
2
2
2
1
2
2
2
2
1
2
1
2
1
2
2
1
2
1
1
2
2
1
2
N
G (91J+I+J+1)
G (91J+I+J-1)
G(91J+I+J-3)
G(91J+I+J-2)
.
68
where
A - L +1M - T
and:
(3.1Ob)
IL, the stiffness matrix, results from leakage; I,
results from absorption;
matrix; and B is
T is
the mass matrix,
the group-to-group scattering transfer
the fission source production matrix.
number of unknowns in each of the G groups, IL and 1
If
N is
the
are GxN block
diagonal matrices composed of G NxN symmetric matrices
IL
g
and D1
g
of the
form
L = Diag[ IL -1
LGI
24 = Diag[ I,9--,
IGI
T and B are in
and
symmetric matrices
general full block matrices composed of G
and Bgg , respectively.
Tgg
is permitted, T becomes lower block triangular;
g
NxN
If only down-scattering
Tgg, = 0
whenever
> g.
A direct method such as the Gauss elimination method [42],[55]
is inefficient compared to iterative schemes for solving large systems
of linear equations, such as Equation 3.10a.
One very commonly used
method in reactor physics calculation is the source iterative scheme
[5],
[56]-[57].
In this method, the equation for the j-th iterative
solution of Equation 3.10a is set in the following form
(
where
+
ghe
)P
=
1B
G [ T ,
g
gg
g
,1 P
gg, -g
=1,2-,G
69
(3.11)
g' < g
l -1;
g'
j
and P(
= L
case, K
g
scheme
1,2,...,
=
, zCl
P
-2
= Col [
_=
g
+]M
> g,
,---,2G
-C ]
is an initial guess.
In our
is positive definite and we can use the Cholesky
g
[58]-[59], which always gives a unique factorization of1K
in
the form
]K
(3.12)
=EET
wherelE is a lower triangular matrix.
LetIK
= (k
), IE = (e
).
Then
we note that according to Equation 3.12 we have
2
k
= e
k
= e
jj
ji
Therefore, e
2 + -+ e.
j2
+ el2eJ2 + --
e
+ e
e
j <i
,
can be determined using the algorithm,
J-1
e
+ e 2
jJ
=[k
-
e
ij
n
2 1/2
]
j-1
ei
= [kij -
(3.13)
einejn ]/ejj
n=1
The matrices]Eand E
T
possess the same band structure asIK
the Cholesky scheme, the numerical inversion of K
g
is
g
.
By using
simplified, and it
requires only a forward and a backward sweep to invert JE and ET
respectively.
In the eigenvalue problem defined by Equation 3.9, only the largest
70
of interest because it
eigenvalue X is
corresponds to the neutron
multiplication constant in the reactor physics.
The largest eigenvalue
of Equaion 3.9 can be determined by the power method
is
Suppose P'(j+l) is
-g
determined briefly below.
15],
[56],
[60] which
the (j+1)-th iterative
solution to Equation 3.11 for group g before renormalization and PCI) is
iterative solution fn-o
the j-th
group g after normalization,
then
the largest eigenvalue and its eigenfunction are defined by
(j+1)l
X(j+1)
)
-
P~j l
--
(3.14a)
-g
S-g
S(j+)
X(j+l)
-
Steps defined by Equations 3.11 and 3.14 are repeated until the following
convergence criteria are satisfied:
(j)
(j+1)
10
max
n,g
()
()
I
.,(j)
P (j+1)
ng
ng
(j)
I
ng
(3.15a)
-X
< E
(3.15b)
~P
The whole procedure described in this section can be best
illustrated in Figure 3.2.
71
Start
j=o
Read
or Generate
adX(0)
PP,(0) andXA
Calculate (IL +IM
)l
for all g using
g g
Cholesky matrix fac-totization method
j+l
j-
Compare P (j+1) by
Eq. 3.11
Compute X (j+1) and P(J+l)
using Eqs. 3.14
Is
(j+1)
_(j)
<No
X(j)Yes
Stop
Figure 3.2
Solution Scheme for A P
Source Iterative Method.
72
=
-B
P Using
Numerical Results
3.4
In this section, some of the numerical results for stationary
eigenvalue problems are presented in order to demonstrate the accuracy
of the variational method using cubic Hermite basis functions as flux
approximations with discontinuous Fick's law current trial
functions.
No special computer programs- were written for the calculations and
each individual case was treated as a single computer problem.
two-group parameters used in
The
the following examples are given in
Table 3.2.
One-dimensional, Two-group, Single Region Problem
Example 3.1
We consider an eigenvalue problem for a one-dimensional twogroup,
is
one region simple diffusion problem.
depcited in
The reactor configuration
Figure 3.3.
The analytical solution for this simple problem can be found
easily and
is
given by
= C sin( ' z)
(z)
(3.16)
z)
$2(z) = sin
where L = 60 cm is the width of the reactor and C is the fast to thermal
flux ratio which is a constand independent of position z given by
2
C =
2
+
Z2
= 3.337902595
21
The largest eigenvalue A for the problem can be shown to be
73
Table 3.2
Two-group Nuclear Constants
(a) Thermal Group
Water Reflectors
Ftil
D2
0.4
0.15
E2
0.2
0.02
0.218
0.0
VEf
2
X2
0.0
(b)
Fast Group
Water Reflectors
Fuel
D
E21
VE f
Xi
1.5
1.2
0.0623
0.101
0.06
0.1
0.0
0.0
1.0
74
-ff2)
[D(j) 2 +
(121)=
]+i 2
1.031303490
2 +12
In Table 3.3(a), comparisons are made for eigenvalues and average
fast to thermal flux ratios obtained by various nesh spacings.
It
is
seen that both the eigenvalues and flux ratios converge to the true
analytical values with decreasing mesh spacings.
The average fast to
thermal flux ratio is defined as the average value obtained by dividing
the fast group result by the corresponding thermal group result for each
space unknowns.
In Table 3.3(b), the thermal fluxes at various points
of the reactor are compared and it is seen that the flux shape converges
rapidly to the analytical shape as the mesh sizes are refined.
The
relative error is defined as
Average Error -Approximate Value - True Value
True Value
(3.17)
In Table 3.3(c), the thermal currents at various points of the
reactor are compared with the analytical results.
It is seen that the
"best" results for current obtained using a variational method that
permits discontinuities in current are not continuous at mesh points.
As the mesh size shrinks, more and more current discontinuities are
introduced, although the magnitude of discontinuities seems to become
smaller and smaller.
Since the problem is a homogeneous one, the converges to the true
eigenvalue and true flux and current shapes are quite fast.
Accuracies
of 10~9 in eigenvalue, 10-5 in flux shape and 10-3 in current shape can
75
Table 3.3
Results of One-dimensional, Two-group,
One-region Problem:
(a)
Eigenval
Example 3.1
es and Average Fast to Thermal Flux
Ratios
Eigenvalue,
Az
X
C
Flux Ratio,
1.031303490
3.337902595
L/2
1.031303421
3.337902612
L/3
1.031303485
3.337902596
1/6
1.031303490
3.337902595
L/12
1.031303490
3.337902595
Analytical
$ =
01
F u1 l1
,
dz
L = 60
Figure 3.3
cm
Reactor Configuration for Example 3.1
76
= 0
Table 3.3 (Continued)
(b)
z
Thermal Fluxes with Relative Error
Analytical
Az = L/2
(error x 104)
Az = L/3
(error x 105)
Az = L/6
(error x 106)
0.0
0.0
0.0
0.0
0.0
5.0
0.13052619
0.13053365
(+0.572)
0.13051992
(-4.80)
0.13052531
(-6.74)
10.0
0.25881905
0.25871065
(-4.19)
0.25879147
(-10.7)
0.25881793
(-4.33)
15.0
0.38268343
0.38248222
(-5.26)
0.38265889
(-6.41)
0.38268085
(-6.74)
20.0
0.5
0.49979957
(-4.01)
0.49996639
(-6.72)
0.49999785
(-4.30)
25.0
0.60876143
0.60861390
(-3.98)
0.60872601
(-5.82)
0.60875732
(-6.75)
30.0
0.70710678
0.70687644
(-3.26)
0.70703144
(-10.7)
0.70710374
(-4.30)
35.0
0.79335334
0.79328492
0.79330454
0.79334798
(-6.15)
(-0.862)
(-6.76)
40.0
0.86602540
0.86568184
(-3.97)
0.86596717
(-6.72)
0.86602167
(-4.31)
45.0
0.92387953
0.92330950
(-6.17)
0.92383484
(-4.84)
0.92387329
(-6.75)
50.0
0.96592583
0.96541020
(-5.34)
0.96580584
(-12.4)
0.96592166
(-4.32)
55.0
0.99144486
0.99122627
(-2.20)
0.99136572
(-7.98)
0.99143708
(-7.85)
60.0
1.0
L0
1.0
1.0
77
Table 3.3
(c)
*
**
z
(Concluded)
Thermal Currents and Relative Errors
Currents are in the unit of D2 and since D2 is a constant in this example, it is just a
scaling factor
Wherever a double-value appears, the upper one corresponds to the value at the left-hand
side and the lower one corresponds to the value at the right-hand side of zk'
Analytical
Az = L/2 (error x 10 3
Az = L/3(error x 10)
Az = L/6(error x 10
)
0.0
1.0
1.00098816(+0.988)
1.00019725(+0.197)
1.00001242 (+0.124)
5.0
0.99144486
0.99081050(-0.640)
0.99125542(-0.191)
0.99143956(-0.0535)
10.0
0.96592583
0.96498125(-0.978)
0.96584471(-0.0840)
0.96589942(-0.273)
0.96597621 (+0.522)
15.0
0.92387953
0.92350042(-0.410)
0.92396515(+0.0927)
0.92387459(-0.0535)
20.0
0.86602540
0.86636800(+0.396)
0.86561671(-0.472)
0.86677571(+0.866)
0.86596197(-0.732)
0.86611033(+0.981)
25.0
0.79335334
0.79358399(+0.291)
0.79293956(-0.522)
0.79334909(-0.0536)
30.0
0.70710678
0.70514840(-2.77)
0.71170298(+6.50)
0.70704737(-0.0840)
0.70701064(-1.36)
0.70722045(+1.61)
35.0
0.60876143
0.60755666(-1.98)
0.60909913(+0.555)
0.60875816(-0.0537)
40.0
0.5
0.49762201(-4.76)
0.49909484(-1.81)
0.50148711(+2.97)
0.49987771(-2.45)
0.50013468(+2.69)
45.0
0.38268343
0.38189902(-2.05)
0.38201065(-1.76)
0.38268138(-0.0536)
50.0
0.25881905
0.26038768(+6.06)
0.25860398(-0.831)
0.25867894(-5.41)
0.25901496(+7.57)
55.0
0.13052619
0.13308801(+19.6)
0.13126710(+5.68)
0.13050076 (-1.95)
60.0
0.0
0.0
0.0
0.0
be obtained even when a mesh size of Az'= L/6 = 10 cm is
used.
This
is not the case when the reactor becomes more heterogeneous, as can
be seen from the next example.
Example 3.2
One-dimensional, Two-group, Two-region Problem
The reactor configuration is shown in Figure 3.4.
of a water reflector region and a fuel region.
The analytical solution
though not easy.
of this problem is also possible [61],
It consists
The largest
eigenvalue is X = 1.020902463 and the flux and current shapes are
combinations of hyperbolic functions as well as trigonometric functions.
Table 3.4(a) gives the eigenvalues obtained by various methods.
It is seen that all eigenvalues converge to the true value and that the
rate of convergence of the present method is comparable to that of cubic
Hermite method (m = 2).
This result should be expected since the only
difference between these two methods is in the initial assumption about
current coefficients in
functions.
the flux trial
In Table 3.4(b),
the thermal fluxes at 5 cm intervals using different mesh spacings are
compared with the analytical solution.
The neutron fluxes are seen
to converge to the true solution, though not as fast as in the
homogeneous case.
quite good.
And when Az = L/12 spacing is
used, the result is
Table 3.4(c) compares thermal neutron currents predicted
by the present method against analytical results.
It is observed that
the most severe discontinuity usually occurs at the material interface
and that the magnitudes of
discontinuities decrease as Az decreases
and as the distances from the interface become larger.
79
Table 3.4
One-dimensional, Two-group, Two-region
Eigenvalue Problem:
(a)
Example 3.2
Eigenvalue, A
Hermite Method*
m = 1
r = 2
Present Method
(m = 2)
Finite
Difference
L/3
1.02483992
1.02145224
1.021534466
1.02397928
L/6
1.02257106
1.02096418
1.020964354
1.02410016
L/12
1.02147201
1.02090512
1.020904977
1.02139257
*From C.M. Kang [24].
0 1 Reflector
0
Figure 3.4
I-
Nclear Fuel
L=60dz
L=6Ocm
L/3
Reactor Configuration for Example 3.2
80
Table 3.4 (Continued)
(b)
z
Analytical
(cm)
Thermal.Fluxes with Relative Errors
Az = L/3
2
(error x 10)
Az = L/6
(error x 10
Az = L/12
(error x 101)
0.0
0.0
0.0
0.0
0.0
5.0
0.08486903
-0.02365409
(-128.0)
0.08575762
(+10.5)
0.08492856
(+7.01)
10.0
0.30882356
0.046933375
(+52.0)
0.32170626
(+41.7)
0.30907204
(+8.05)
15.0
0.83938228
0.87290130
0.90204065
0.84008346
(+3.99)
(+74.6)
(+8.35)
20.0
0.57723662
0.58098635
(+0.650)
0.57136244
(-10.2)
0.57652361
(-12.4)
25.0
0.41559201
0.42719308
(+2.79)
0.39992663
(-37.7)
0.41503756
(-13.3)
30.0
0.55164109
0.53331112
(-3.32)
0.54655308
(-9.22)
0.55164881
(+0.140)
35.0
0.68057223
0.71250788
(+4.69)
0.68137634
(+1.18)
0.68059766
(+0.374)
40.0
0.79133441
0.77795075
(-1.69)
0.79187389
(+0.682)
0.79135182
(+0.220)
45.0
0.88074847
0.88394510
(+0.363)
0.88098813
(+0.272)
0.88075832
(+0.112)
50.0
0.94639693
0.95235240
(+0.629)
0.94650201
(+0.111)
0.94640103
(+0.0433)
55.0
0.98650822
0.98907119
(+0.260)
0.98651891
(+0.0108)
0.98650875
(+0.00537)
60.0
1.0
1.0
1.0
1.0
Normalized to $2 (L)
- 1.0
81
0110
000
(C6C*O-)6T8ZCZVOO-0-
(V6TO*O-)V698VZVOO*O-
( T6 80 0 0-) CL9 Z'700 " 0-
(9ZZOOOO+)VT96VZVOO@O-
(6VZ*O-)0VSVTZOO-0-
(TTZO go+) z LZVSTZOO " 0(OTEO 0 o-) T5;TE5;Tzoo go-
0*0
(ZS*T-)LSZLZSTOOOO(06Z*O-)S6C9ZTi700s0-
000
ST9C9TZOO*OST96'7Z'70O 00-
0
0
01,09
M; TOO *0-) LS VO CZ900 " 0-
(C9TO*O-)6696ZZ900*0(moo 0o-) I L TE vo goo 00(SETO *0-) 9 TE Z170 800 00-
(T9*Z+)'7ZC0TOTO*O(C&9T-)SZ696Zi700*O+
(OCV*0-)!9L9SOO9OO "0(T9 *Z+) SL09VZOOO *0-
(LO*T+)6T'7L6S90000-
(ETZ*O-)C9ZLTZ900-0-
T99OCZ900*0IM700800"0-
0 0s;i7
oeov
(T6ZOOO*O+)SC8LC9600*0(05;'70 *0-) 9 LVEE9600 P0(VZT*O+)99TBZ60TO00(CUT-)EST69LOT000-
0Z0Z+)C69CO6CTO*O-
Q 0TT+) Z6TLCTZTO 00(SOT-)ZOV989000'0+
(i7L*Z+)L899LZZTOOO-
QVT 00-) T89EZ9600 00-
OTSLE9600*0SSSVT60TO*O-
069C
0
(9C *tt+) TTV9ZOOTO so-
(E*LC-)99LO90ZSO*O+
U *ZT-)06ZTOVZLO 00+
(L6*L-)Zl7f7O9C9LO@O+
(TSZ+)6C6TTSOTO'O+
('7*i7Z-)866LSOZOOOO+
(O-EZ+)OEC6TSTTO-0-
(C*99-)6L66CZCOOeO-
(6 -6Z-) Z9068TOSO '0+
(CCT-)96ZSZ9COO-O+
(oi7o'7+)STZgCZTTO*O-
(V9*C-)ZCLO9COTO*0(9T'70 *0+) V9989 LOTO *0-
('76'6+)0099f7OTZO-0-
(6*5;Z-)ZZLC'8T'7TO*0(LC6C+)S8C89L6TO*O-
(99Z-)LO6V69900'O+
(C'7 "8+) L96CS9 COO *0-
(6C*6-)Z608900000(C*ZC+)9S90S9TTO*O+
(99 *9-)9TTZ89LZO *0+
(96*9-)8OC0SZ9Z0*0+
(9*CT+)OL8666LOO*O+
('79*L-)OOT90000*0(TE *9+) P79809TO *0-
6TZ80960000V68T9LZOO*O+
0
9
c*4
00
COOLLM060+
OZL9L6CCO*OSTOV17T6TO 006LT'79LOT000-
u
0
Q VS *0+) VCO L 8 LCOO '0-
z OT x aolaa)g/rl = ZV
z OT x a0alq)ZT/q = ZV
(T*CT+)i7SL9ZZZOO-0-
(69L*0+)9VCC86TOOs0-
(TB*z+)6VTZLSEOO*O-
(ZOV "0-)09ZTSLCOO *0-
(L*9T+)LTCOLOOTO*O(8&CS-)CLS6UVTO*O+
(OT x J0alq)C/rj = ZV
VTV99LEOO*OOZZ896TOO*O-
0
0
T-e:)-F:jATvUV
saolig 9AT:IL-Tall q:jTm s:luaaanD 7[uma9ql (0)
i7eC ;aTqLl
(p;apnT,3uoo)
Example 3.3
Two-dimensional,
One-group,
One-region Model Problem
An eigenvalue problem for a two-dimensional neutron diffusion
equation is considered in this example.
The configuration consists of
uniform nuclear fuel and is shown in Figure 3.5.
D,
1.
For the one-group
and \vif, we use the thermal group constants given in Table 3.2.
For the present method, we only calculate the eigenvalue and flux
shape using a mesh size of L/2 = 20 cm in both directions because of
large number of unknowns (In fact, the number of unknowns is 38 per
group for a 2x2 problem, from Table 3.1).
This eigenvalue is listed in
Table 3.5 with eigenvalues for different meshes obtained by the cubic
finite element method and the finite difference scheme,
Note that the
present method yields a worse eigenvalue than the finite difference
scheme using the same mesh size.
The flux shapes at y = 0 cm and y
=
20
cm obtained from present method are compared with the true cosine shapes
in Figure 3.6.
The currents are not continuous across all mesh
interfaces.
Example 3.4
Two-dimensional,
Two-group,
Two-region Problem
In this example, we consider an eigenvalue problem of the twogroup neutron diffusion equations.
The system consists of a fuel
region inside and a reflector region outside (Figure 3.7).
In Table 3.6, comparisons are made for eigenvalues obtained by
various methods.
For cubic (m = 2) Hermite method, the results given
correspond to using Equation 3.5 as flux trial functions and setting
83
Table 3.5
Eigenvalue 1/X of a Two-dimensional,
One-group, One-region Problem:
Example 3.3
Analytical, 1/X = 0.9230903697
Ar
Cubic Heri_e*
Finite Difference
Present Method
L/2
0.9230904055
0.92280573
0.9212300089
L/4
0.9230903703
0.92301801
L/6
0.9230903697
0.92305812
* From C.M.Kang [24].
= 0
dY
0
L
Nuclear
Fuel
L
~=
# = 0
0
* = 0
L = 40 cm
Figure 3.5
Reactor Configuration for Example 3.3
84
1.0
0.8
0.6
0.4
0.2
0.0
0
30.0
20.0
10.0
40.0
Flux at y = 20.0 cm
Analytic
0.9230903697
2x2
0.92122300089 (-0.2023%)
1.0
0.8
0.6
0.4
0.2
0.0
0
Flux at y - 0.0 cm
Figure 3.6
Flux Shapes:
85
Example 3.3
1)
ID1P (01)(1)
=
E)2 P (91)(2)
=
JD3P (1)(3)
= 3D4 P (1'1) (4)
(3.18)
whereIDk is the diffusion constant matrix at k-th corner of point (X.,X.),
at singular points (D
2)
Eq.
3.18,
1
D3
ID2 )4 ), and
ElDP (l9(+) =
at all regular points (1
2 P('t(-),
1 2ili ~j
1
D3 =ID2 D4 ),
and ]D2 P (0 '1 (+) =I E
2i~j3
i9j
before applying the weighted
residual process in Ritz-Galerkin method.
It is seen that both the cubic
Hermite method and the present method for Ar = L/2 yield accuracy
comparable to that of the finite difference scheme for Ar = L/20.
In Figure 3.8, thermal fluxes at y = 0 cm and y = 20 cm for (i)
Cubic Hermite method, Ar = L/2, (ii) Present variational method,
Ar = L/2, and (iii) Finite difference method, Ar = L/20 are compared.
Note that present method gives slightly better flux shapes than the
cubic Hermite method.
From the results of these numerical examples, it is observed that
the present variational method using discontinuous Fick's law current
trial functions yields eigenvalues and flux shapes comparable to those
obtained from the cubic Hermite method and is generally better than
the linear Hermite method and finite difference method.
However,
since the number of unknowns in present method is considerably larger
than that of the cubic Hermite method if the same mesh spacing is used,
we conclude that the cubic Hermite method is better than our present
method.
86
Eigenvalues 1/X of Two-dimensional,
Table 3.6
Two-group, Two-region Problem:
Example 3.4
Hermite Method*
Ar
*
m = 1
m
=
2
Present
Finite
Method
Difference
L/2
1.0802150
1.1082321
L/4
1.0962251
1.1134916
1.0797120
L/6
1.1040456
1.1140943
1.1105031
1.0783013
1.1075521
From C.M. Kang [24].
i =0
dY
L/2
L
0
Fuel
# = 0
dX
L/2
Reflector
L
# = 0
L = 40 cm
Figure 3.7
Reactor Configuration for Example 3.4
87
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
20
10
0
30
40 cm
Thermal Flux at y = 20.0 cm
Kang's,
0 0 0 0
1.1082321
L/2
Variational, L/2
1.1075521
Finite Difference, L/20
1.1105031
1.6
1.2
0.8
0.4
0.0
-0.4
0
10
20
30
40 cm
Thermal Flux at y = 0.0 cm
Figure 3.8
Thermal Flux Shapes:
88
Example 3.4
CHAPTER 4
FINITE ELEMENT SYNTHESIS APPROXIMATION METHOD
IN NEUTRON DIFFUSION PROBLEMS
As described in the beginning of this thesis, the finite element
methods have been shown [24],
[62] to approximate accurately flux
solutions and eigenvalues of multigroup diffusion theory when applied
to problems having homogeneous material within the mesh regions.
How-
ever, when this method is applied to a reactor system having very
complex geometrical details,
the region mesh sizes must be limited
unless some types of homogenization procedures are used.
If the mesh
spacing Js chosen such that some mesh regions are heterogeneous, then
application of the variational principle given in Chapter 2 results in
weight averaging of the nuclear constants with products-of the basis
functions and their derivatives, as given by the approximate trial
functions.
A useful homogenization procedure which is commonly used in
reactor physics analysis is to homogenized the nuclear material within
each mesh region by flux weighting with an assumed flux shape determined
a priori within that region in order to preserve reaction rates,
n (r) (r)dr
<
'1
$n (r)dr
in
89
where $n (r) is the assumed flux shape in region n.
In large reactors the core is usually composed of a lattice of
heterogeneous fuel subassemblies containing fuel, clad, coolant
channels, and possibly control rods or coolant-filled control rod
channels.
Each subassembly can be divided into several distinct
homogeneous regions whose microcell macroscopic group, constants are
found by multigroup energy-dependent calculations [63].
Detailed
subassembly solutions are then found for each subassembly by assuming
that the current on the outside boundary of the subassembly is zero.
Flux weighting the nuclear material in each subassembly with the
corresponding detailed subassembly solution for each subassembly
region (according to Equation 4.1) then results in regional homogeneous
nuclear constants which may better approximate the physics of the
region.
Proper use of detailed flux weighted constants can lead to accurate
criticality measurements, but the detailed fine flux structure within
each region is lost since it appears only in crosssections homogenization
and not in the approximation.
The present finite element synthesis
method is intended to solve this difficulty by combining the detailed
subassembly solutions with finite element basis functions in the flux
approximation, so that the fine structure can be retained in the flux
solution.
The application of this method to 1-D problems has been
shown to be successful
[30].
The purpose of this chapter is
to extend
the finite element synthesis method to 2-D diffusion problems using
bl-linear basis functions.
90
4.1
Derivation of the
Difference Equations
4.1.1 Linear Basis Function Approximation in
1-D
The proposed finite element synthesis method, utilizing linear
basis functions and defined as nonzero within each mesh region k,
k
=
1 to K in one-dimension, is given by
+ ( 1 (1)
kN1()(-
Uk(z) = $k
Uk(z) = $P(x)[$*k
Vk(z) = nk ()
k-0(lx
k +*k
$
Vf(z)
=
ip(x)[$
+ 1
LIDk
where:
-k~
+ P
(4.2b)
k+1
k+11
(1)
x
(4.2c)
k1b
1 -* -l)O~"*
()P]
(4.2d)
is the unknown approximate group flux column vector at point
P
zk, and *',
k9
(0)(1-x)P*
(4. 2a)
(1xP k+1I
+k-
[k(O(1)k
Vk(z)f=lk
D
+ l kk+l
P*
-k(1) x
+
(0)(l-x)P
X Pk1
k9
*k' kk , and nk are GxG diagonal matrices composed of detailed
group flux %g9k(z) and group current ng,k(z) solutions, and their adjoints,
defined as nonzero only within region k.
Because of the variable
transformation between z and x, $k (0) represents $Pk(zk), and $ k(1)
represents ik(zk+l); neither of which for the moment is allowed to
be zero for any region.
The detailed current solutions are given from
91
the detailed flux solutions by Fick's law as
(4.3a)
r (z) = -IDk (Z)
diP*(z)
rJ(z) = + IDk(z) k
(4.3b)
k
dz
As a result, the current trial functions are related to the flux trial
functions by analogous expressions.
Also, be definition of the detailed
flux solutions,
rOk)
=
k(1) = rT(O) = rik(l) = 0
From the forms of the trial functions, we see that the flux
continuity conditions are obeyed since
Uk(0) = Uk-(1)
U*(0)
= U*k(1)
(4.4a)
P
(4.4b)
P*
The current trial functions, however, are discontinuous.
It is evident
that this approximation reduces to the simple linear basis function
finite element method if detailed flux solutions for each group are
taken to be constant.
The detailed derivation and the resulting
difference equations using approximation Equations 4.2 are given in
Chapter 3 and Appendix C of Bailey's thesis
4.1.2
[30].
Bi-linear Basis Function Approximation in
2-D
The flux trial functions for the finite element synthesis method
92
using bi-linear basis functions and defined as nonzero within each
mesh region (i,j), i = 1 to I, j
1 to J in two-dimension, is given
=
by
U
(x,y) = $
(xy) [$
+
U* .(xy)
i9j
=
(1,0)P2
1 (0,1)P
+
$
i91iq
(0,0)P
1
i+l,
(X)P2 (Y)P1
(x,y)[$*
1
+
piqj(l9)
2
2
]
i+1 ,j+1
(4.5a)
(0,O)P (x)P
1
1
yPt
i9j
+ +$ *10P() 1 YP i+1,j+ + J1 (1"1)P (X)P (Y)P*~
q (,)2
11)P2
2
i+1,9j +1
(01)P (
+
2
,j+1]
(4.5b)
where
P1 (x) = 1
x
are the univariate linear basis functions.
P
tion group flux column vector at point (Xi,Y ),
is the unknown approximaand $x,
$*
are GxG
diagonal matrices composed of the detailed group flux solutions
$gqi
(XY)and their adjoints, respectively, defined as nonzero only
within region (ij).
For the moment, we assume that
is nonzero
at the region boundary.
The detailed current solutions are given from the detailed flux
93
solutions by Fick's law as
(xy)
-(x,y)
=-
*
a~ ;
i
qy
*i~
(XY) =j-.-D
s
t
1a
(x,y)
hyj j
(x,y) =
(x,y)
a~
x19
-
-
=
$
(x,y)
hxi
(<x,y)
I*
(xy)
D
ay
iD (x,y)
$4)* (x,y)
(x,y)
yj
and (*
where E
solutions and n
solutions.
are X-direction detailed current and adjoint current
and rn*
are Y-direction current and adjoint current
Then the X-direction and Y-direction current trial functions
which obey Fick's law are given by
Q
(xY)= E
(xY)[~1 (0,0)P0(x)P
+ (1 0)P2
+
+
+ $
p (Y)P
Pi
(11)P
2
Md d
iJ (0,1)Pl(x)p2 (y)Pj+1]
D,
(x,y)i
.(x,y)[1j-(0,O)qlpx)
+ $i2i(1,0)q 2 (x)P1 (Y)p+
+ $j<0,)0
1x)P
2
(Y)p
1
1
,
j+ i
l
94
()P&
1y
(1,1)q2
2i+1j+1
(4.6a)
R 91(x,y)
s
=n
XYlijqo
1()
+ lp-1 (19O)P (X)P (Y)Pi~.
+ i(O
,1)
Yp19
+ 4-1 (191)p (x)P (Y)P+ji
P(X) P(Y) p
l1i.(x,y) [*-1 (090)p (x)iql(y)pij
yj
-1
1O
+
(qlyP+,
-1
(,)
xq(~
(4. 6b)
Q*
(xy)
=
C
xy0-1
2 (x) 1 (
+
+
h1 I ~
x-i
00pW
+ * 1 (191)-P (x)P
Pi~
1 9j
2 9o~
isj
(x,y) 0
(0 O)ql(x)Pl(y)P*
(XY) 10-1
191
isi
-1
+* 19, ~
ii .(,i
Y*
Jp()P
oo)I
i
Y*
2
(x)Pl(Y)p*+s
+ 4* -1(0,1)q (X)P (Y)P*
95
+
+~
12
1IqW
YP
2i+lj+
l
(4.6c)
R*
(Xy)
=Ti*
(XY)[1
)*
(0,0(l.))Px
yj
1
+2
+
P*
2
(0,1)P
i+1 ,j +
* 1 1 )P2
2
i +1,j+1
, J(4.6d)
1
where
2
q10
1
q (x)
= -
12
Tq
P1 (x)
dx
d p
a-P(x)
=
w~
+1
=-1
We note that the flux trial functions are continuous only at the
mesh points and are discontinuous at the interfaces of mesh regions in
general.
The current trial functions are discontinuous also across
region interfaces.
Also, be definition of the detailed subassembly
solutions
TI
(0,
=
(x, 0)
=
=
ni
(x,1)
=
(
,y) =*
n* (x,0)
96
=
(1,y) = 0
n* (x,1)
=
0
Since the flux is not continuous across region interfaces while
Fick's law is still valid within regions, the suitable variation
equation to be used is Equation 2.15.
For two-dimensional case,
Equation 2.15 can be written in the form
I
J
l
h xi
xil h
=1j=1
+
0
D
I-1
dy[6U*
I
h
dy[I6U T
T(x,y) Ai(xY)u
(x,Y)
,3
'
*i
(x,y)Q
~rjU
(x,y)-SR*T(xy) ]D_
(1,y)-6U*+1,
i+lj
T
(x,y)R
(x,yJ
i++l oj(0,y)+Q 1 9 (1y)]
J-1
=1
l
h
dx[6U*
(x,1)-6U*
11i~~
0io
1=1 j=1
+ -1h
dy [6Q*+1j
i=1 j=1
J+
J1
- 6QT (xy)
ii
+
dx
f0
j+1 (xO)] [R
~
(O,y)+ Q* (,y)] I[U
(xO)+R
(xl)]
(1,y)
(0,y)-U
0
1
I h
2i=1 j=1
dx[6R*,,j+1(x,O)+
0ioii~+
6
R*
(1,y) ]T [U
(x,1) ]=0
+(x,0)-U
9
(4.7)
Substitution of Equations 4.5 and 4.6 results in the equation
I+1 J+l
i=l
T
p
+ab
+ a
j=1
+ba
-iP
P
+bbP
iP-i,j
+
Pi-1,j+1
+iccPj+ i+1
=0
P
+ cc
P
+ cb
P
+ ca
Z-i 9j i+l Ij z-i~j i+llj+11
z-iqj i+lsj-l
97
(4.8)
where "undefined" quantities (corresponding to points that would lie
outside the physical limits of the reactor) are always set equal to
zero.
,---,
, a
The GxG matrix coefficients {a
c
lare
integral quantities having the mathematical form A - 1B and
are defined in detail in Appendix E.
Zero flux boundary conditions are easily imposed.
For example,
if zero flux conditions are imposed on the left boundary of the
reactor, we can set P
.
= 0, j = 1 to J + 1.
This also requires
j = 1 to J + 1 be zero, which in turn requires the 6P
that P*,
coefficients in Equation 4.8 to vanish.
If all four sides of the
reactor have zero flux conditions, then allowing arbitrary variations
,
in P*
j = 2 to J, results in a system of (I-1)(J-1) equations in
(I-1)(J-1) unknowns.
Zero current boundary conditions are found
using symmetry considerations.
If, for example, zero current
boundary conditions are imposed on the right boundary, then "boundary
condition equations" can be derived by assuming pseudo-regions
and height h
(I+lj), j = 1 to J of width h
image properties of region (I,j), j
ID
's having mirror
1 to J, about line X=XI+1
=
(4-9a)
.+j
(x,y) =1 19 (1-x~y)
1+1,3I,j
;
/AI+1j
=
j
=
1 to J
(4.9b)
Ij (1-x,y)
and the detailed flux and current solutions for these regions are
assumed to be symmetric and anti-symmetric in
the X-direction,
respectively, to the corresponding detailed flux and current solutions
98
in
the regions (I,j),
j = 1 to J:
UI+ 1 ,j(xY) = U
(1-x,y)
(4.10a)
U+1 j (XY)= U*
(1-x,y)
(4.10b)
II~j
j - 1 to J
V I+
(x,y) =-V
(4.1c)
VI+1,(xY) = -V*
j(-xy)
(4.10d)
(1-x,y)
The addition of pseudo-regions (I+1,j), j = 1 to J to the summation
in Equation 4.7 results in the calculation of coefficients jaa
(j=2 to J+1),
bb
wI+l 9j
! 1 + 1,j (j=l to J+1)) .1 +,j(=l
(j=l to J+1), and bc
to J),
aI+ 1 ,j(j=2 to J+1),
(j=1 to J) in Equation 4.8.
-i+l 2j
If all
four sides of the reactor have symmetry conditions, then allowing
arbitrary variations in P*,
i=1 to I+1, j=1 to J+l, results in
a system of (I+1)(J+l) equations in the same number of unknowns.
Like the situation in one-dimensional case [30], a serious
drawback of the approximation given by Equations 4.5 and 4.6 is
that
it does not allow the use of detailed flux solutions containing
explicitly zero flux boundary conditions.
However, such detailed
solutions can be allowed by modifying the trial function forms in
the boundary regions.
For example, if a zero-flux condition is
i~(x,y) is
imposed on the left boundary and a detailed solutions *
given such that $4~(0,y) = 0 in the region (1,j) for a particular
J, the trial functions of Equations 4.5 and 4.6 can be modified for
99
this region as
U
(,
1,))
=
+
9 $ (,)2
2,j+1)
it
i~i
(x,y)
U*
+ =
Q* (x,y)
+
=
*(x,y)[$*
2P
(4.lla)
29
1
(1,0)P (y)P*
*l*
(1,1)P
2 (Y)P
1 (x~[44
-1j+1
10
(4. 1b)
)P,
(4.llc)
Q
(xy)
(x,y)[p* (1,)P (y)P
=
(4.11d)
R
= n[(x
,(XY)
+
)
1I)
2
(1,)P
xy 0 xy(-
+
P1 (11)p2
(Y)P
2 j+1
(1,0)q nN2,j
(~2jl
19
+
*
(1,1)q2
,j1I
100
(4.lle)
R*
(x, y)
= T*
(x,y) [$*
1
(1, 0)P (YP
*
i2j
hyJ
.j 1,j
+$*
-
1,3
-1
(1,21)P2()P2,j+
1,0)q (y)P*
1
2,3
2,j 1 1
(1,1)q 2 (
(4.1f)
The above equations are equivalent to fixing the X-direction basis
functions for this particular region (1,j) so that
P1 (x)
=
ql(x)
=
1
, P2 X
0
,
q2 (x)
=
in Equations 4.5 and 4.6.
0
(4.12)
In this way, the imposed zero boundary con-
dition is explicitly given by * 13(x,y) rather than by the form of the
trial functions.
The use of these special trial functions in the
boundary regions alters the definitions of some of the matrix coefficients,
but the whole system of equations are of the same form as that of the
ordinary zero-flux boundary conditions.
Regardless of the types of boundary conditions imposed, Equation
4.8 results in an NxN matrix problem of the form
1
A P =
(4.13)
1 P
where A and B are independent of A.
The order N of the matrix equations,
which depends on the chosen boundary conditions, is given in Table 4.1.
Careful examination of the matrix coefficients given in Appendix
E shows that neither A nor B are symmetric matrices in general.
101
Only
Table 4.1
Matrix Order N of the Bi-linear Finite
Element Synthesis Approximations as a
Function of the Imposed Boundary
Conditions.
1 - Zero Flux
2 - Symmetry
Boundary Condition-Type,
Matrix Order
N
Left
Right
Top
Bottom
1
1
1
1
G(I-1)(J-1)
1
1
1
G(I-1)J
1
1
2
2
1
1
2
1
GI(J-1)
2
1
1
1
1
1
1
2
2
G(I-1)(J+l)
1
2
1
2
1
2
2
2
1
1
1
2
2
1
2
1
2
2
1
1
1
2
2
2
2
1
2
2
2
2
1
2
2
2
2
1"
G(I+1)J
2
2
2
2
G(I+1)(J+l)
GIJ
G(I+1)(J-1)
GI(J+l)
102
when the geometry of the problem is symmetric about the 450 line, do
A and lB become symmetric.
4.2
Calculational and Programming Techniques
The matrix equations which result from the approximations given
in this chapter are of the form of Equation 4.13, which is identical
to Equation 3.9 in the previous chapter except that A and B are not
symmetric in the present situation.
The conventional group indexing
followed by spatial indexing within each group in ordering the unknowns is assumed.
For the double spatial indexing, we shall order
the unknowns first by Y-direction indexing, j, followed by Xdirection indexing,
i,
for each
j.
Because of the complexity of the matrix coefficients (aa
ab,
---
,
cc
} encountered in the approximation,
it
is
difficult
to find a systematic way for assembling the coefficients so that a
single computer program can be used to find the eigenvalue X and
corresponding eigenfunction directly when the geometry and regional
detailed nuclear constants are the input.
Therefore, the calculations
were performed in four steps as follows:
(i)
Calculation of the Detailed Subassembly Solutions:
Calculation of the two-dimensional subassembly detailed fluxes
and adjoint solutions were performed using the existing code PDQ-7
[64]-[66].
The PDQ-7 program solves neutron diffusion-depletion
problems in one, two, or three dimensions in rectangular, cylindrical
103
Up to five energy groups,
or hexagonal geometries.
overlapping thermal groups, are permitted.
including two
For the present applica-
tion, the non-depleting option for two dimensions, rectangular
geometry, one or two energy groups was needed.
The boundary conditions imposed were zero-current conditions
on all four boundaries, in accord with the definition of detailed
Also, subassembly (i,j) is divided into MxN
subassembly solutions.
homogeneous subregions having widths of z
x,m
in the X and Y directions.
and z
y,n
, respectively,
Omitting group subscripts, the detailed
flux solutions for each group in subassembly (i,j)
is
represented
by a set of (M+1)(N+l) points
(11.)
Calcul a
(4.14)
(xy) = {$mn ; m=1,M+l, n=1,N+l}
$
o'f
tio
'Dou. -b IC
-- A
Tnt-ygrals :
i-neer
The matrix elements required for use in the approximation
methods are combinations of various double and linear integrals of
products of subassembly detailed solutions and polynomial functions.
The double integrals are calculated for each subassembly (ij)
from
the basic integral unit
DIA
dx
=
ij
0
J
dy f
(x,y)C. .(x,y)x
.(x,y)g
y
(4.15)
0
where the functions f .(x,y)
i,3j
and g
.
(x,y) represent flux and/or X-
or Y-direction current solutions for the same or different groups.
C
it
. (x,y) represents a group nuclear constant which is homogeneous
in each subregion (m,n) of the subassembly (i,j)
104
C
(x,y) = {Cm,n ; m=1,M, n=1,N}
(4.16)
and k and k are positive integers in the ranges 0 < kk < 2.
If we assume that for the detailed flux solutions of the form
Equation 4.14, the average flux in subregion (m,n) is given by
m,n
m,n +$+,n
4
lPm+1n+l + $mn+l
+
4.17a)
the average X-direction current in subregion (m,n) is given by
D
M,n ~
mn 1'-P
2
m+1l,n
2x9m
l(41b 'Pm,n+l)]
+
m,n)+('m+1,n+l
(4.17b)
and the average Y-direction current in subregion (m,n) is given by
D
m,n
m,n+1 ~
2z
'Pn
+(
l ,n+l
-
$m+ln)]
(4.17c)
y,n
then the basic integral unit can be broken into sums over each sub-region (mn)
M
N
m1 n
DIA
k+l
(xm+1
m ,n mn m,n (k+l)
k+l
m
P+l
Yn+l ~
(Z+l)
k+l
n
(4.18)
The linear integrals are integrations along the subassembly
interfaces.
For a horizontal interface between subassemblies (i,j)
and (i,j+l),
the line integrals can be calculated from basic integral
unit of the form
SIA =
J
dx f j(xl)gij+1(x,0)D 9(x,l)xk
105
(4.19)
(x,l) is the detailed flux solution at the bottom edge of
where f
the subassembly (i,j), gi 1j+1 (xO) is the detailed flux soltuion at
(x,1) is the diffusion
the top edge of the subassembly (i,j+l), D
constant at the bottom edge of the subassembly (i,j) and 0 < k < 2.
If we assume that the average fluxes along this particular
interface are given by
2
mN+l
1 (
m, 1
m,N+l + $m+,N+l sub.(ij)
(4.20a)
+
12+
(4.20b)
m+l,l sub.(i,j+l)
then Equation 4.19 can be written as
M
DIA = m=
1
k+1
m,N+m,lD m,N (k+l) (xm+ 1
k+1
m )
(4.21)
Programs DOB1 and DOB2 which calculate quantities F,
(a,b;c,d),
defined by Equation El and consisting of combinations of various
double integrals, in one-group and two-group, respectively, are
listed in Appendix F.
Programs LIN1 and LIN2 which calculate various
linear integrals in one-group and two-group, respectively, are also
listed in Appendix F.
(iii)
Computation of Matrix Elements:
This step involves the calculation of matrix elements from the
results of the previous step.
Careful bookkeeping is necessary in
order to insure the accuracy of the computation.
However, for a
reactor system consisting of a relatively small number (< 4) of
106
different subassemblies,
this is
not a difficult task; it
usually
takes several hours of work.
(iv)
Calculation of Eigenvalue and Eigenfunction:
The source iterative scheme and power method discussed in
Section 3.3 are used to find the largest eigenvalue X and corresponding
eigenfunction once the matrix elements for L , Mg , T
known.
The inverting procedures for K
changed because K
Cholesky scheme of
[42],
is not symmetric.
g
,
and B
, are
IL +Mg , however, must be
g
g
Thus, instead of using the
matrix factorization, the Gauss-Jordan method
[55] was used to invert Y .
g
Programs MAN1 and MAN2 which calculate the eigenvalue X and
its corresponding eigenfunction for one-group and two-group problems,
are listed in Appendix F.
4.3
Numerical Results
In this section we present the results of three cases obtained
through the use of the finite element synthesis method.
also analyzed by the
Each case was
pure linear finite element method using detailed
flux weighted nuclear constants.
These results are compared with
reference solutions obtained from the finite difference code CITATION
[67].
4.3.1
Case 1:
25 Subassembly Reactor Configuration Made up of 2
Different Types of Subassemblies - One Group.
107
The reactor configuration and its 2 different types of subThe one-group nuclear constants
assemblies are depicted in Figure 4.1.
of subassemblies are given in Table 4.2.
The detailed flux solutions
for each subassembly were found using the finite difference code PDQ-7
with symmetry boundary condiAons and a 20x20-mesh region per subassembly geometry as
indicated in Figure 4.2.
The resulting detailed
flux solutions for Sub. 1 and Sub. 2 are shown in Figures 4.3 and 4.4,
for y = 0 cm and y = 4 cm, respectively.
homogenized, subassembly
The detailed-flux-weighted,
nuclear constants are given in Table 4.3,
and are used in the linear finite element calculations.
The reference solution for this problem was found using the same
mesh intervals in each subassembly as in the calculation for detailed
subassembly solutions.
With the symmetry about X=20 cm taken into
account, this is a 50x100-mesh region problems for CITATION.
The graphical results for the one-group approximation methods
for this case are shown in Figures 4.5, 4.6 and 4.7 for three
elevations Y = 12 cm, 20 cm and 24 cm, respectively.
The flux dis-
continuity across subassembly boundaries, which is inherent in the
approximation, is shown in Figure 4.6, and more clearly in Figure
4.7.
The magnitude of the gap along a boundary depends on those two
It
bordering subassembly detailed flux solutions at that boundary.
is seen that the present
synthesis method using a coarse mesh of
8 cm gives reasonably good results compared to the reference solutions.
They are farther better than finite element method using the same
108
Subassembly 1
Subassembly 2
-4 cm
4 cu-
1 cm
G---
4 cm
8 cm
I
-W
-m
1
8 cm
1
1cm4
cm
8 c
2
1
2
1
Y = 12 cm-
Y = 20 cm
10 cm
Y = 24 cm-
1
2
1
2
_
__
1
.
2
.1
4U rm
WC
Figure 4.1
__
A 25 Subassembly Reactor
0*11
Configuration and
Its 2 Different Types of Subassemblies Case 1
109
s
Table 4.2
One-group Nuclear Constants of Subassemblies
of Figure 4.1 - Case 1
Subassembly 1
Material 1 (fuel)
I (cm~ 1 )
Material 2 (moderator)
0.0232927
0.0
0.04
0.015
D(cm)
0.3
0.1
IV
2.5
a (cm~1)
Subassembly 2
Material 1 (fuel)
Material 2(absorber)
J (cm~1 )
0.0232927
0.0
la (cm~1 )
0.04
0.500
D(cm)
0.3
0.5
v
2.5
Table 4.3
Homogenized Subassembly One-group Nuclear Constants
- Case 1
110
A
8 cm
8 cm
Symmetric Partitioning in Both X- and Y-directions:
1(1.0cm) + 4(0.5) + 3(0.25) + 1(0.15) + 1(0.1)
Figure 4.2
Mesh Geometry in
a Subassembly --
111
Cases 1 and 2
1.0
Sub.
1
0.95
0
N
0.90
r-4
0.85
Sub.
2
0080
0.754
0
Figure 4.3
2
4
x(cm)
= 0 cm
Detailed Flux Solutions at y
Case 1
112
8
6
-
1.0
Sub. 1
0.8
0.6
N
0
z
0.4
Sub.
2
0.2
0.0
0
Figure 4.4
2
4
8
6
x(cm)
= 4 cm
Detailed Flux Solutions at y
113
-
Case 1
0.35
0.30
o
o
o
o
Linear FES
0.25
x 0.20
o
0.15
/
0.10
0
O
0
0.05
/
0
000
00
/o
/0000
0.00
_
_
_
__
16
8
0
x (cm)
Figure 4.5
Flux at Y = 12 cm
114
-
Case 1
20
0.9
0.8
-
-
-
Linear FEM
00
(
0.6
/o/
E
0/
of
/0
/.4
0/
0.40
0
0.2
0.0t
16
8
0
x (cm)
Figure 4.6
Flux at Y = 20 cm
115
-
Case 1
20
1.0
Reference
Linear FEM
0.9
e
0
0
0 Above
*
*
eBelow
Linear
FES
//
'nr
0
/
0.8
0)0
0
0
0.60
0.5
8
10
Figure 4.7
12
14
Flux at Y = 24 cm - Case 1
116
16
coarse mesh.
Comparisons
of eigenvalues, X, obtained from different methods
are shown in Table 4.8 (p.146).
We see that the present synthesis
method gives a much better eigenvalue than the linear finite element
method using identical coarse mesh.
4.3.2
Case 2:
25 Subassembly Reactor Configuration Made up of
One Type of Core Subassembly and One Type of
Reflector Subassembly - One Group
The difference between case (1) and case (2) is that in (1)
fuel is present right up to the edge of the reactor, while in (2)
there is a reflector zone of moderator subassemblies.
The reactor
configuration and its two different types of subassemblies for case
(2) are depicted in Figure 4.8.
The one-group nuclear constants of
subassemblies are given in Table 4.4.
The detailed flux solution for
Sub. 1, obtained by using PDQ-7 with geometry indicated in Figure 4.2,
is shown in Figure 4.9 for y = 0 and 4 cm.
The detailed flux weighted
homogenized nuclear constants, used in finite element calculation, are
given in Table 4.5.
Note that for reflector subassembly, we assume that
the detailed flux solution is constant over the whole subassembly,
because there is no fission inside.
The resultant flux weighted
homogenized constants are thus the same constants as for the reflector
subassembly itself.
The reference solution for this case was found using the same
mesh intervals in each subassembly as in the calculation for detailed
117
Subassembly 2
8 cm
10
1
8 cm
Y =
8 cm
2
---
Y = 12 cm
-'
Y = 20 cm
-.
2
2
?
40 cm
2
1
1
cmQ
.40
I.
Figure 4.8
1
2
,q
A 25 Subassembly Symmetric Reactor Configuration and Its 2 Different Types of Subassemblies - Case 2
118
Table 4.4
One-group Nuclear Constants of Subassemblies of
Figure 4.8 - Case 2
Subassembly 1
Material 1 (fuel)
)
Material 2 (moderator)
0.0228646
0.0
a.(em~)
0.045
0.015
D(cm)
0.4
0.4
f(cm~
2.5
Subassembly 2 (reflector)
Table 4.5
Homogenized Subassembly One-group Nuclear Constants
- Case 2
Sub.
D(cm)
Ja(cm~)
Vf (cm~)
1
Sub.
0.6
0.4
0.04179954
0.02
0.05106339
0.0
119
2
1.0
0.98
0.96
y= 4 cm
0.9/
0.92
0.90
0
2
4
6
8
x(cm)
Figure 4.9
Detailed Flux Solutions for Subassembly 1
--
Case 2
120
0.8
0.6
rZ4
0
z
0.4
0.2
0.0
0
8
Figure 4.10
16
Flux at Y = 12 cm - Case 2
121
20
1.0
0.8
0
0.
00
j:0.6
0.2
d
e
dReferen,
0 0
0.0
--
Linear
0
o
Linear
1
0
8
16
x(cm)
Figure 4.11 Flux at Y = 20 cm - Case 2
122
0.35
0.30
/
/
S0.25
00
0.20
Reference
0.15
-
-
oo
--o
Linear FEM
o Above
Linear
Below
BelowFES
0.10
8
10
12
14
x(cm)
Figure 4.12
Flux at Y =8 cm - Case 2
123
16
subassembly solutions.
Because of the quarter-core symmetry,
the
problem size was 50x50.
The graphical results for this case are shown in Figures 4.10,
4.11 and 4.12 for elevations Y=12 cm, 20 cm, and 8 cm, respectively.
The magnitude of flux discontinuities across the interfaces of two
different types of subassemblies is much less than that of case'l.
This is because there is no strong absorber present in the reactor
core and thus the subassembly 1 detailed flux shape is
flat.
relatively
Also, because of this fact, the results obtained from the
linear finite element method and linear synthesis method are closed
to each other and both predict quite good flux shapes and eigenvalues
(Table 4.8) compared to the reference solutions.
4.3.3
Case 3:
49 Subassembly Reactor Configuration Made up of 2
Types of Core Subassemblies and One Type of Reflector
Subassembly - Two Groups
The reactor system for this problem consists of a 25-subassembly
core made up of 2 different types of subassemblies surrounded by a ring
of 24 identical water reflector subassemblies.
The reactor configuration
and its 3 different types of subassemblies are depicted in Figure 4.13.
The two-group nuclear constants for three different kinds of materials,
(fuel, absorber and water moderator) which form the three types of
subassemblies,
are given in Table 4.6.
The detailed subassembly flux and adjoint flux solutions- are
found using PDQ-7 with a 14x14-mesh region per subassembly.
124
The
partitioning of mesh intervals is
X and Y directions.
4(1.5cm) + 6(1.0) + 4(1.5) in
Figures 4.14 - 4.17 illustrate these subassembly
detailed solutions for Sub. 1 and Sub. 2.
Sub.
3,
both
For water subassembly,
the detailed solutions are constant over the whole region.
The homogenized two-group constants for each type of subassembly are
given in Table 4.7.
The reference solution for this case was found using the same
nesh intervals in each subassembly as in the calculation for detailed
subassembly solutions.
The total number of mesh regions
is 49x49,
because of quarter-core symmetry.
The graphical results for this two-group problem are shown in
Figures 4.18 - 4.21.
It is seen that the coarse-mesh linear finite
element method cannot predict the thermal flux peaks and dips in the
reactor.
This indicates that finer meshes must be used to obtain good
results.
The linear synthesis method, whether calculated by adjoint
weighting or flux weighting, gives reasonably good general shapes both
for fast and thermal fluxes
in the core region.
However, the magnitudes
of thermal peaks in subassemblies of type 2 can be very different compared to those of reference solutions.
The cause of this phenomenon
is the inability to predict the thermal flux peak in
the reflector
near the core-reflector interface by using flat flux shapes in the
reflector subassembly and treating each subassembly as one mesh-region.
One way to overcome this difficulty is to put more mesh points
in the reflector region by partitioning the reflector subassemblies
further.
Another way is to
use some prescribed flux shapes in the
125
Subassembly 2
Subassembly 1
9 cm
2 cm
K -
18
1118cm--
cm
A
9 cm.
2 cm/
cm
18 cm
9 cm,
2 cm
0
-
Subassembly 3
>
--,8
~~~~~18 cm
cm
I I
I
Y=27 C
3
1
2
3
2
12
1
2
1
1
?2
3
Y=63 cm
126 cm
-!
3
1
2
1
2
1
3
3
2
1
21
1
2
3
3
1
2
1
2
1
3
3 ,1
3 ,1
3 ,A
'
)
3
X
Figure 4.13
-
126 cm
A 49 Subassembly Symmetric Reactor Configuration
and Its 3 Different Types of Subassemblies - Case 3
126
Table 4.6
Two-group Nuclear Constants of Three Different
Materials in Subassemblies of Figure 4.13 Case 3
f21
fuel
absorber
Material 1
Material 2
moderator
Material 3
1.436
1.092
1.545
0.02647
0.003185
0.028824
0.007293
0.0
0.0
0.01596
0.0
0.02838
1.0
xl
D2
0.3868
0.3507
0.3126
2
0.1018
0.4021
0.008736
2
0.1531
0.0
0.0
0.0
-
-
v f
X2
Table 4.7
Homogenized Subassembly Two-group Nuclear
Constants - Case 3
Sub.1
Sub.
2
Sub.
3
1.4030165
1.4464270
1.545
0.02423738
0.009547083
0.028824
0.006593731
0.006596349
0.0
0.01442972
0.01714810
0.02838
D2
0.3854548
0.3749848
0.3126
2
0.1129898
0.08698106
0.008736
Vyf2
0.1473952
0.1287213
0.0
D
v
127
1.0
0.9
y=9cm
0. 9(
\
/
0.9
0.92
0.9C
0.0
4.5
Figure 4.14
9.0
13.5
Sub-
1
Sub.
2
18.0
x(cm)
Detailed Fast Fluxes for Subassemblies - Case 3
128
0.5
Sub.
1
Sub.
2
0.4
/
y =9cn
0.3
r/
0.2
0
0.1
y =9 cm
0.0
0.0
4.5
9.0
13.5
18.0
x(cm)
Figure 4.15
Detailed Thermal Fluxes for Subassemblies
--
Case 3
129
0.85
y-9 cm
0.80
-
-~~
y= 0
0
-
-
cm
0.75
0
z
0.70
y=cm
0.65
Sub.
1
Sub.
2
0.60
0.0
4.5
9.0
13.5
18.0
x (cm)
Figure 4.16
Detailed Fast Adjoint Fluxes for Subassemblies - Case 3
130
y = 0 cm
~
~
1. 0~
y
=9 cm
0.8
r-4
P4
S0.6
*r4y
y=9
0
z
cm
0.4
0.2
Sub.2
0.0
0.0
4.5
9.0
13.5
18.0
x (cm)
Figure 4.17
Detailed Thermal Adjoint Flux for Subassemblies - Case 3
131
.7
-
0 0
/
P6 00
'-0.6
*
e
0.4
0/d
e
0
0a00
e
E
0/0
o//
e
0/
0/
0
oq
I0.
50
90
Linear FEM
Q10
/0
0 000o
Adjoint
o o oeeo
Flux
10.0
-
,
0.0
IE
54.0
36.0
18.0
ic (cm)
Figure 4.18
Fast Flux at Y
132
=
27 cm
-
Case 3
63.0
---
1.0
0
o
0.8
/n
0.
po
/
0.6
r-/
0~
/
0o
/0
nz/ 0/0
Reference
0.4
o/
o/0-
Linear FEM
0/0
e-
-
0
o4o
04
o o Adjoint
FES
46
0.0
-LiearF2
0
36.0
18.0
x
Figure 4.19
54.0
(cm)
Fast Flux at Y = 63 cm
133
Flux
-
Case 3
63.0
.35
0.2
x*
H
0
0
(
on
e0
N
00
u
0.1
0.
0
.
0 oo0
0/
00
*/
0
0
0
4
03.
.
0
(
40
0
O W
0
00
____
0.0
18.0
___
___
___
___
36.0
Thermal Flux at Y = 27 cm
134
___
54.0
x (cm)
Figure 4.20
____
-
Case 3
___
___
___0.0
63.0
0.5
o o
o o
0.4
Adjoint
Flu FES
0.3
og
0/
o
0.1
o
0/
.0
0.0
18.0
36.0
54.0
x (cm)
Figure 4.21
Thermal Flux at Y
135
=
63 cm
-
Case 3
63.0
reflector subassemblies
instead of flat
shapes.
Figure 4.22 shows the average X-direction flux shapes for water
subassemblies at the left side of the reactor, found from the reference
solution.
The Y-direction flux shapes are assumed to be flat.
These
fast and thermal flux shapes d1 ffer from those obtained by a simple
1-D calculation involving only one 18-cm region of water and one 18-cm
region of fuel:
0-flux I
0
Wnter
ThO1
18
I 0-current
36 cm
by 2% at most, using the same mesh intervals (14) in each region.
Therefore, in the actual calculation where we normally would not have
the reference solution in advance we can obtain the flux shapes from
some simple auxiliary calculations.
The flux shapes for water
subassemblies at other three sides of the reactor can be found by
symmetry considerations.
Figure 4.23 shows the flux shapes for the top-left corner water
subassembly, again from the reference solution.
Simple auxiliary
2-D calculation involving 3 water regions and 1 pure fuel region:
136
0-flux
Water
Water
0-flux
36 cm
18
0
0-current
18
Fuel
Water
36 cm
0-current
shows that these flux shapes differ only slightly (<3%)
with each
other.
For the modified calculation which follows below, we shall
use Figures 4.22 and 4.23 as the regular flux as well as adjoint
flux shapes for various water subassemblies.
The use of the flux shapes shown in Figures 4.22 and 4.23 is not
legitimate in our present approximation, since these shapes do not
How-
have zero normal currents on all four subassembly boundaries.
ever, we can show that for
the present case, use of these flux
shapes merely adds some leakage terms to the calculations of those
matrix elements (Appendix E) which correspond to reflector-core
interfaces.
These terms are of the
form
137
1.6
1.4
1.2
1.0
Thermal
PL4
0.8
0
0.6
0.4
ast
0.2
0.0
4.5
9.0
10.0
18.0
13.5
x (cm)
Figure 4.22
Flux Shapes for the Water Subassemblies at the
Left Side Used in Modified Calculation
138
-
Case 3
1.2
1.0
0.8
0.6
o
Thermal
z
0.4
Fast
0.2
--
0.0
--
'
4.5
0-,
9.0
13.5
0.0
18.0
x (cm)
Figure 4.23
Flux Shapes for the Top-left Corner Water Subassembly
Used in Modified Calculation - Case 3
139
+
+
2
0,1)$*T
(1 dyP (y)P (y)[$*T
0
-
m
@
n
(1 0)
2,
2,j
(
0ly)c
J
(1,y)$~1
and where m ant
(1,0)
, )$ 2 , (0 y)* 2 ,j
for the interface between subassemblies (1,j) and (2,j)
ticular j,
1,j
, are either 1 or 2.
for a par-
The two terms in
the square brackets are of the same sign.
maximum absolute value for the
A rough estimate, using the
first term and minimum absolute value for the second term, gives a
largest value of
18 1
2
- (0.0111)(0.981)]
5
;[(1.003)(0.0111)
= 0.00073
for the fast leakage and a largest value of
18 1
2
-
[(1.053)(0.00433)
for the thermal leakage.
-
(0.00433)(0.936)]
= 0.00152
These values are less than one tenth of one
percent of the original leakage matrix elements.
The results shown in
the following pages are the results obtained by ignoring these
additional small terms.
The results obtained by using these modified shapes for reflector
subassemblies along with the previous shapes for core subassemblies
are shown in Figures 4.24 - 4.27.
It is seen that the flux shapes,
especially the thermal flux shapes, improve significantly in the core
region (X > 18 cm).
The eigenvalues obtained from various methods are
140
0.7
0.6
0
-O
0"@
0.4
Q)
IN
0.3
0
0.2
Reference
-
----
Linear FEM
o 0 0 O Adjoint
0.1
FES
* *
*@eFlux
S0.0
0.0
18.0
36.0
54.0
x (cm)
Figure 4.24
Fast Flux at Y
141
=
27 cm-
Case 3, Modified
63.0
.0
.8
/
.6
/
/
8
O
0.4
Reference
-
-
- -Linear
FEM
-.
00 0 0 Adj oint
/
8
FES
eee
(fi~
C.0
a
eFlux
0
18 .0
36.0
54.0
x (cm)
Figure 4.25
Fast Flux at Y =63 cm -Case
142
3, Modified
6.
.25
-
--
-
-
-
Linear FEM
Adjoint*
o o00 o oa0.20
FES
e*e.e
e Flux
0.15
H
o
0
z
loow
eo
0
0.10
00.0
/
0
'00
0.0
18.0
36.0
54.0
x (cm)
Figure 4.26
Thermal Flux at Y = 27 cm
143
-
Case 3, Modified
63.0
0.5
0.4
o Adjoint1
o o o
FES
Flux
0
0
x
0
0.3
0
rX4
"cd
0.2
o
/
u
0.1
0.2
0e
0.0
54.0
36.0
18.0
x (cm)
Figure 4.27
Thermal Flux at Y
144
=
63 cm
-
Case 3, Modified
63.0
compared in Table 4.9.
It is observed that:
(i) the linear FEM
using flux-weighted constants gives better results than the linear
synthesis method and (ii) flux-weighted results are better than
adjoint-weighted results
in linear synthesis method.
Although
there is some mathematical reason to expect that the adjointweighted result should be better than the flux-weighted result,
there is no proof of this expectation and it is common that in many
numerical problems the reverse is true (as in this case).
From the
above example, we conclude that the method of
synthesizing detailed subassembly solutions with finite element
basis functions in two-dimensional diffusion problems is quite
encouraging.
Thus, instead of the calculating of a large, fine-
mesh problem to obtain the eigenvalue and detailed flux shape
throughout a geometrically complicated core, we can solve the
same problem by first determining a few, much smaller, fine-mesh
subassembly solutions and then performing a very small coarse-mesh
calculation.
145
Table 4.8
One-group Eigenvalue A Obtained from Different
Methods - Cases 1 and 2
Case 2
Case 1
Reference
1.0081406
1.0071173
Linear FEM
0.9447535
0.9918033
(-6.29%)
(-1.52%)
1.0017971
(-0.63%)
0.9919463
(-1.51%)
Linear Synth.
Table 4.9
Two-group Eigenvalue X Obtained from Different
Methods - Case 3
Case 3
1.0428543
Reference
1.0397139 (-0.30%)
Linear FEM
Linear
Synth.
Adjoint
1.0593464
Weighited
(+1.58%)
Flux
1.0510194
Weighted (+0.78%)
Unmodified
146
1.0319060
(-1.05%)
1.0380392
(-0.46%)
Modified
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
5.1
Conclusions of the Study
Our proposed approximation, which couples detailed subassembly
flux solutions together with bi-linear finite element basis functions,
has a mathematical structure that is very similar to a coarse mesh
bi-linear finite element approximation in which detailed flux shapes
have been used beforehand to flux-weight the nuclear constants in
each region.
However, the two methods are conceptually different and
become equivalent only when all of the coarse mesh regions are homogeneous.
The proposed method is also similar to existing synthesis
methods which use detailed flux solutions or other known flux shapes
directly in the trial function for the flux.
in that it does not require
However, it is different
full core, detailed flux shapes as input
but instead stitches together a few fine-mesh subassembly solutions.
The numerical results indicate that the proposed scheme is
able to predict criticality accurately as well as the detailed flux
shapes for each group.
The results indicate that although both the
proposed scheme and the finite element method with flux-weighted
constants give good criticality estimates, the actual detailed flux
behavior is much better approximated by the proposed method than
the finite element method using coarse meshes.
As in any coarse mesh approximation method, inaccurate results
147
can occur when the coarse mesh region sizes chosen are too large.
Thus it
would be useful if
there were error bounds that could be
ascribed to the finite-element synthesis method.
The accuracy of
finite element methods is known to improve geometrically as the mesh
sizes is decreased.
Thus there is a useful error criteria for the
finite element methods.
On the other hand, the inability to predict
error estimates has always been a major drawback of synthesis
techniques, even though through proper physical insight and experience, accurate results can be obtained.
Whether or not error
bounds can be found for solutions obtained by combining synthesis and
finite element flux shapes is not yet known.
Another thing which is lacking in this study is the computation
time comparison between various methods.
inability
This is because of our
to generate computer code for which the eigenvalue and
flux shapes of a problem can be obtained directly without any
intermediate matrix elements calculations, and we have to obtain the
results step by step.
In conjunction with the study, we found some interesting results
related to the finite element Hermite basis functions approximations:
(i) Under certain conditions described in the end of Chapter 2,
Fick's law is a natural consequence of using the variational
functional
F3 .
(ii) The solution that is found by applying a variational
method using continuous cubic flux trial functions and
148
discontinuous Fick's law current trial functions is not
necessarily the best.
We found in Chapter 3 that the cubic
Hermite method, which defines continuous flux trial functions
in such a way that the current trial functions are also continuous throughout the problem domain except at singular
points, yields more accurate results even though the trial
functions space is more limited.
5.2
Recommendations for Future Study
For future study of the finite element synthesis method, some
areas which deserve closer attention are:
Development of error bounds for the finite element-
i)
synthesis method.
The close similarity between the present method and
the finite element methods may allow an extension or generalization of
error estimates previously developed for finite element methods.
ii) Examination of the matrix properties of the synthesis
method.
This is necessary in order to guarantee convergence to a
positive eigenvalue and an everywhere positive flux solution.
The bi-linear finite element synthesis method can be extended
to three-dimensional diffusion problems as well as to a higher degree
of Hermite basis polynomials.
The mathematical principles are the
same as for the present method, and the main concern would be the
algebraic complexity of handling large numbers of lengthy equations.
149
Also possible is
forms in
the extension of using the proposed trial
function
spatial overlapping synthesis methods for multi-dimensional
reactor problems, where the discontinuities of flux and current
trial functions occur at different places.
150
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155
BIOGRAPHICAL NOTE
The author, Shi-tien Yang, was born on July 9, 1946 in the
city of Hsinchu, Hsinchu County, Taiwan.
He received elementary school
education in his home town and then graduated from Taiwan Provincial
Chien-kuo High School in Taipei, in July 1964.
He then attended Tunghai University in October 1964, without
taking the usual requirements of College Entrance Examination because
of his good academic performance in high school.
In June 1968, he
received his Bachelor of Science Degree in Physics.
During his
four years' stay in Tunghai University, he was honored as an outstanding
student every year, and was given the highest honor for a student with
his name engraved on the wall of the University Auditorium upon his
graduation.
in
He was also elected to the Phi Tau Phi Honorary Society
his senior year.
After one-year mandatory service in the Chinese Air Force, he
came to the United States in August 1969 and enrolled at the Massachusetts
Institute of Technology in February, 1970.
The author completed this
work and obtained the degree of Doctor of Philosophy in 1975.
The author is married to the former Yu-fen Lai and is expecting
a baby in August 1975.
156
APPENDIX A
TABLE OF SYMBOLS
Energy group index which runs from the highest to the
g
lowest energy group as g - 1 to G.
The position vector.
r
Scalar neutron flux in energy group g at r
9(r)
2
(neutrons/cm -sec).
Neutron current vector in energy group g at r
(neutrons/cm -sec).
Diffusion coefficient for neutrons in energy group g
D (r)
at r (cm).
Macroscopic total neutron removal cross section in energy
Z(r)
group g at r (cm~).
~ (r)
Macroscopic neutron scattering cross section from energy
-1
group g' to energy group g at r (cm
).
The eigenvalue of the diffusion problem.
Xg
Fission spectrum yield in energy group g.
V1
Macroscopic fission neutron production cross section
(r)
-1
in energy group g at r (cm
D(r),
J*(r)
)
!
Scalar group flux column vector of length G and the
corresponding adjoint flux vector.
J (r), J*(r)
Vector group current column vector of length G and the
corresponding adjoint current vector.
157
ID(r)
GxG diagonal group diffusion coefficient matrix.
1(r)
GxG diagonal group removal cross section matrix.
W(r)
GxG group scattering out cross section matrix.
F(r)
GxG group fission-production cross section matrix.
A(r)
GxG group removal, scattering, and production matrix.
k
Ofte-dimensional spatial index which runs from the leftmost first region to the right-most K-th region, as
k = 1 to K.
i,j
Two-dimensional spatial indices.
For horizontal direc-
tion, i runs from the left-most first region to the
right-most I-th region as i
-
1 to I and for vertical
direction, j runs from the top first region to the
bottom J-th region as j = 1 to J.
Unit vectors in- the X and Y directions in
z
a 2-D problem.
The one-dimensional axis variable divided into K regions
such that each region k is bounded by nodes zk and
zk+l
X, Y
The two-dimensional axis variables divided into I and J
intervals, respectively, such that each region (ij)
is bounded by lines X = Xi, X = X+1' Y = Yj, and
Y = Yj+1'
x
A dimensionless variable defined in
each region k as
x = (z-zk)/(zk+1-zk), such that 0 < x < 1 as
z
< z < Zk+1.
It is also used as the dimensionless
158
variable in X-direction in a 2-D problem.
A dimensionless variable in Y-direction in a 2-D problem,
y
defined in the same way as x.
U(r),U*(r)
Scalar group flux and adjoint (or weighting) flux trijl
function (column vectors of length G).
V(r),V*(r)
Vector group current and adjoint (or weighting) current
trial function (column vectors of length G).
Qi' ,Qi *
X-direction group current and adjoint current trial
function (column vectors of length G in region (i,j)).
R
,R.
i,j'
i,j
*
Y-direction group current and adjoint current trial
function (column vectors of length G in region (i,j)).
k(z), k*(z) Detailed one-dimensional subassembly flux and weighting
flux solutions in coarse mesh region k.
k (z),nk*(z) Detailed one-dimensional subassembly current and weighting current solutions in coarse mesh region k.
i
Detailed two-dimensional subassembly flux and weighting
4,9*
flux solutions in coarse
*
mesh region (i,j).
Detailed two-dimensional subassembly X-direction
current and weighting current solutions in coarse mesh
region (i,j).
i
two-dimensional subassembly Y-direction current
*
,9
IDetailed
and weighting current solutions in coarse mesh region
(i,j).
159
A
Discretized matrix form of the GxG group diffusion,
absorption, and scattering matrices.
B
Discretized matrix form of the GxG group fissionproduction matrix.
P
The unknown approximate group flux solution vector which
may contain group current unknowns.
160
APPENDIX B
INNER PRODUCTS FOR HERMITE BASIS FUNCTIONS
Nonvanishing inner products for the univariate basis functions
uP (x)
k
for m = 1,2 (as defined by Eqs.
1.10 and 1.11) are listed
below:
(i)
m=1
0-
uk-1)
0+
(uk
0+
uk )
d
,
d
0+
(x
1
d
-
0+
( dx% P duk)
(ii)
d
1
-
=
uk
d
1
3
0+
=-1
dxk-)
(- -u
0uk
0uk+l
Uk
0+
0+
(uk
1
0+
dO0uk t1k)
(
( x
0-
1
0+
(uk
u
0-
(
0+
0uk , Uk+
d
(t
Uk
d
= 1
0-
Uk~,
1
6
0-
0+
dx uk
d
-1
) -
-
1
2
0-
x
= 1
d
0-
dx
kl
m = 2
0(uk
0+
uk-1
00(uk ,uk)
= 70
01+
(uk , uk-)
13
35
01(uk , uk
9
161
13
420
11
210
0+
(uk
0+
Uk
13
35
0-
9
0+
(uk , Uk-1
1(uk
210
1+
0+
(uk , uk
d
(uk
d
(
Uk)k-1
0-
0-
0+
0+
UkUk~
d
(d
d
'
0+
d
Uk
1-
1-
1(uk
uk
0+
Uk-1
1
105
1+
1
uk
105
1+
1uk ,uk+1
d
1
(
2
0,
u-
.__1_
140
1+
1
k-1
10
:
d
1+
1
0+ u1-
.1
10
0+
S2
d
(t
1
~
2
Uk ~ k+)
d 1-
1
-
d-g Uk
0
d
0-
(x Uk Uk
1-
(uk
0-
uk ' Uk+
Uk
(
1
140
13
420
0- 0+
dx uk
1+
uk-1
1+
11
210
01+
(Uk , Uk+l
40o
(uk
1-
420
11
0uk
13
(uk , Uk+1
13
-
210
1-
0+
0+
1-
11
k , uk
70
uk , uk+1
1+
0+
1-
( -Uk
TO
162
1+1
66
Uk-1)
1-
Uk
=0
0
d
(x
1+
0+
)
Uk
=
d
1+
d
1+
(t
uk
0- d
0+
6
d
uk *dx
uk-1
5
t- uk
0- d
(-uk *d
0uk
6
5
0+ d
uk
d
1- d
(dx Uk
d
1- d
(t Uk
d
7
d
(t
0
'
dx
1+ d
Uk
0-
0+
Uk
1+ d
U~
k
0uk+1
1
Uk
01
dx Uk-i-t
6i
d
6
5
d
1
-1
10
1
d
1+
1
k
l
d
1-
1-
uk
d
1-
(a;
Uk
d
d
7
d
v
d
10-
t
1
1+ d
1+ d
u k v'x
10
1
36
1+
uk-1)
1-
2
ixUk
Uk v
1
10
1uk+1)
(*
10
6
1
1-
Tx
0+
(7Uk
d
16
16
Uk
0+
d
1
,1+
dx uk-1)
(xuk
0+
uk-1
1
0-
d
0
1-
0-
d
=
Uk+1)
(xUk
(t uk ,dx uk
d
1+
(a; uk , uk )
1
1
1+ 0(d
dx uk
uk+1
d
d
1
10
1+
; Uk
1-1
*k+1)
15=
=2
15
3
The inner products for multivariate basis functions can be determined
using the univariate inner products.
163
APPENDIX
C
DIFFERENCE EQUATION COEFFICIENTS RESULTING FROM USE OF
THE FINITE ELEMENT APPROXIMATION METHODS IN CHAPTER 2
The GxG matrix coefficients resulting from the linear finite
element approximation and the cubic Hermite finite element approximation in multigroup diffusion theory are defined below in various
The coefficients are given in terms of assumed homogeneous
sections.
regional nuclear constants through the use of the GxG group matrices
]Dk and Ak.
the form A
~1
k ~
Since Ak
-
these coefficients are of
F,
Various inner products of the finite element
1
basis functions used to facilitate the calculation are listed in
Appendix B.
C.1
Coefficients of the Linear Finite Element Method Equations
(as defined by Eqs. 2.20)
Interior Coefficients; k = 2 to K:
ak = 6
bk
ck
& k-l hk-l1
3
k-1 hk-l + "khk) + (Dklhk-1.
k hk
=6
k- /hk-l
~k/k
Symmetry Boundary Condition Coefficients:
b
1
Ah h
3
1
h1
/h
11
164
+Dk/h k)
Ah- Dl/hl = ck (k=l)
c
= ak (k = K+l)
+ 6/hK
K
b K+ 1 bK+l
C.2
1h
=
+
+ 31D. hl
Coefficients of the Cubic Hermite Finite Element Method Equations
(as defined by Eq. 2.38)
Interior Coefficients; k
a2
13
420
k
k
clk
c2 k
a3
k
=
k-1/hk-1
f-1 hk
210
70
+ Ak h)k
2
Ikh
k-k-1
L11
13
420
-1
k-1
Akhk
2 I-1
1
-1
k-1
2
Akhk
1
1
2
-l k-1
165
kIk/hk-1
+
k/hk
khk
2 13
420
-1 l
10
k-1
2
-lhk-1
blk
b2
2 to K:
k-1 h-1-
70
ak
-
1
1
-1
k
k/hk
1
a4k
b3k
6
c3k
c4 k
k-
1
3
k-1
.
-1
1
k-i
3
-l
0hk-1
k-1
-1 2
2
-1
-210 (Dkl hk-12 Akfk hk Ak)
b=11
b4k
-1
13
)DI(h
-1
-1
kkl+hiDk
1
2
kAk+l1
-1
Dh
h3
k
2
1hk (ID~
1 3
-1
b4 1
ID
-1
30 hkM)kl
1 40 IDkk Akmk
Zero Flux Boundary Condition Coefficients:
-1
b471
1
105
c3
1 420
13
D
=1
a3
420 K
K+1
-1
2
1 h ID
-1
2
1
111 h A
c4 1 =.11D
1 140
a4
3
A ID+
h 111
+
10
1)
c 4 k (k
k
1l) Lh ]D'
30 1 1
I
h1 1 IA
1
2
hi
c3 k (k
-
1
0(
a3k k
= -1
140
K+1
2 h-1
.-1
-1 3
14=
KDK
+15
105 DK 'A'K
-1
+ KK 3
13
166
K+l)
h
hKIK
K+1
1)
= a4 k = K+)
k
Symmetry Boundary Condition Coefficients:
bl
=3 A
35,
h +
5 11
cl
=-JA
705
h
c2
- 4h2
1
13
1
K+1
bK+l
-
1
-
1
_
= c2
(k = 1)
(k =1)
6 K/hK = alk (k =K+1)
13
4201K h
1
/h
cl
1D/llk~k1
2 17-
lK+l
alK+1hK
a2
/h
hK
K
DK
167
+
10
= a2
(k = K+)
k (
APPENDIX D
DIFFERENCE EQUATIONS RESULTING FROM USE OF CUBIC FINITE
ELEMENT APPROXIMATIONS WITH DISCONTINUOUS FICK'S LAW
CURRENT TRIAL FUNCTIONS
The difference equaticas resulting from the cubic (and bi-cubic)
finite element approximations with discontinuous Fick's Law current
trial functions in 1-D and 2-D are given below.
The nuclear constants
are assumed to be homogeneous in each region.
D.1
Difference Equations in 1-D (derived from Eq. 3.3)
For simplicity we define
d[a,b]k = dah
k
+ db Dk/hk
and
d[a,b](k,k')
d[a,b]k + d[a,b]k
and with the understanding that all nuclear constants outside the
reactor region should be set equal to zero, we have
(i)
From taking arbitrary variations in all P*, k=1 to K+l:
k
2
-
+
[
1+.3
1 2
I'
k-1
[( ,
k-l +
6] (k-1,k) k +
1
[
1 ]k-1
Sk-l,+
,1]k 4k,+
3P
k-l Qk,- +
k
2 ]k
[
k+1,- = 0
;
168
Pk+l
k
=
1 to K+1
(Dl.1)
(ii)
From taking arbitrary variations in all Q
[1OT
, 1 ]k
k +
1 ]k
-30t,
(iii)
0 42
k
k+l
k = 1 to K
qk+l,- - 0
(Dl. 2)
From taking arbitrary variations in all Q
_,
113
- [ ,
110 21
1
-
_
+ +5 [ ,2k-1
12
(iv)
2 ]kQk,++
5 7
k = 1 to K:
1
k-1
- = 0
4k,-
3
1
;
k-1
k-1,+
k = 2 to K + 1
k
=
2 to K+1:
k-l
-
(Dl.3)
Zero fluxboundary conditions:
Eliminate k = 1 and k = K + 1 equations from Eqs. D1.1
and set P
(v)
1
=
P
K+l
=
0.
Zero current boundary conditions:
Eliminate k = 1 from Eq. Dl.2 and k = K + 1 from Eq. D1.3
and set Ql,+
=
QK+1,-
=
0
169
k
D.2
Difference Equations in 2-D (derived from Eq. 3.81
For simplicity we define
d[a,b,c]
dah
i~j
xi
h
y i.;j
d[a,b,c]
h
h
+ db h
+ dc -
h
d[a,b,c]
Ljj
x
h
D
yj
i~j
+ d[a,b,c]i, 1 ,
With the understanding that all nuclear constants outside the
reactor region should be set equal to zero, we have
(i)
From taking arbitrary variation in all P (O),
i Si
1+1, j=l to J+l:
27
3
(0,0)
1-1, j-1
175 28
1 39
-50[28'
+
+
9
-
13
-
P
]0
[3
)(
(1,1)
01 i-1, j- 1 Pi-1,j-1()
3 [399 2
[4 ,,
1 33
175 2-8'
+
-
13
4 2-00 [:1
+1
(,0)
P il'-
13
[ 9 , -13,
+
1-1,j-1
9
4
61
11
-11
P (0,0)
ij-l
i-,lj-l+j-1)
ij-1
(1,0)
1,j-1
P
170
)(-9
i=l to
+1
1
(i-1,j-1+i,j-1) Pi (091)
1 143
2100 42
13
2
. 1 143 13
2100 42 * 2'
i-i,j-1
+7 [ -1-1]
1
13 -13
4200 429
+
+
1
(1,0)
9-3,
~ Pi+1,j-l
(091)
2, -1]
350 28'
(1,1) (2)
ij-l
Pi+0j
39 9
1 39
+
(1,1)
i,j-1
i,j-1
-1-1
1,j-1
p(1,1)
(2)
i+1,ji-1
,-26 ,9](i-1,j-1+ i-1,j)
13 13
+350 42'
1(1,0)
(i-1, j-1+j-1, j)
, -11,
+
1o-o
1-1, j
(01)
+ -7--[2-81-1 T) -1Ipi-igj
-
1
1
33
9 ,-1
(]9)
143_1 913
+2100T2
1 2
-1,j
p (1,1) (1)
i-1,j1
-171
(+)
-
2100 42
-11
]
*2
1
P(4
1-1,j i-i,j
13 1(00)
78
+57[t=,1,
(l-l+itj-l+1-1,lij
itj
+'
-$11]
P
175 42 '2 *
1 (Ij-l+itj) itj
143
1
13
175 42~'
1
143
[42 ,
+
2
13
-7 [2 ,
2100 1
11
13
(0,1)
21 (1-1, j -l+1, j -1)
11 p(1 1) (1)
91i~j
11
210021'
(1,1)
1 i-1,j
ij"2
11
-
2100 21
3 39
+75 14,*-26,
(1,0)
,
,j-1
(-)
p(0,1)
P,
1 i-1,j+i,j)
1
1, 143
11
(i-1,j-+i-j) ,j
+
(i 1) (3)
(1,1)
1,
(0,0)
i+1,j
(,j-l+ij)
172
1, j
C-)
-
13
13
1
33
(1,0)
1
1
-
[
-1
143
2 itj
[.L
175 '28"
-1
1 '39
350 I- 28P
1 39
350
-1]
9
-13,
413
4205 42' 1
+
3 39
[ 33
(3)
]
(0,0)
i-i
si+l
p (1310)
i-lgj i-lgj+l W
9
-1 ,j
(0,1)
j Pi-1,j+1
(11) (4)
-(2j)
175 1r 4
+
i+1,j
-
i-Iti
131
-
P 3] (101) (2)
ilj-1 P
2
+2100[42
(+)
(0,1)
i+1s
93
1
210542'
+7
MM
4ii
33
-T(T
P(0,1)
9
+175 289
~ ,-6 (i-1,9J+i 9)
i , J+1
9-
(10)H
1
33
-5 [t
9
-1]
P-tj ij+1
173
-1
213
+
1
+10
,j
143 13
[42 *2 *
73
-1
175[t28'
1 39
350 28'
1 [39
350
ij
-1]
i+lj+1
13]s.,
13
9
-13, -
(1,0)
i+1,j+1
p(0,1)
,j
i*j
+ 4130 1
ij+1
P (0*0)
*i,j
9
,
1
(1,1)
i~j+1(3
-1,j
210 0 42 * 2 *
+
ij
i+i,j+1
i 1,j+1(3)
0
i = 1 to I + 1, j = 1 to J + 1
(ii)
From taking arbitrary variations in all
i = 1 to I,
j
=
1 33. 9
175 28'4
+
[.
,t 3,
1 to J + 1:
ij-1
p (090)
1,0(j-1
-2]
174
(D2.1)
P
(+ ,9
143
l
13
(o ,1)
+2100 42 '2
1
13
+lol2'*
1,j-1
~ -'iM-1 1i,j-1
, - 1 3 P1
3 3
1
350 28' 2'
13
1(1,0)
lijj-1
1
13
1400 42
13
9
[143
13
+
+175[42
+ -5~
i+1,j-1
13(01
[
Ps(-
-1]
0[ 42, -1,
+
(+)
(2)
i+,j-1
(0,0)
3(1,j-1+i,j)
3 1
,13,
l
(1,1)
ij-1
'
2
1,j
j)
,j-i
2100'21
11 11
ii
(,1)
1111
21
1
9
11
22
,j-1
11
1
11
~ij
p(l 1) '1)
+1050' -
~ 1050 21'
+
13
, -
+
1,j-1
ji,
22
3' *
(1,1)
jj-
1
,
175
(4)
_9)p(0,0)
13' 13
+ 356[t4Z
3 13
350 [42
(igj-l+1,j)
(1,0)
13
9
1 143
1
(i,.-1+1,j) 1i+1,j
1
+2100 42
1
143
2100 4t
-
((0,1)
3]
2i+1,tj
i
l(0 1)
13 1
1 ij-1
1
11 22
1400 212 9
p(1,1)(3
i+1,
(3)
1 33 9
(0,0)
+ 5[25' -49
1,ig ij+1
3, -2]1
1
143
2100 42
13
P (1,0)
91igj+1
13
2'
9j
13
1
(4)
1,jio+1
(0,0)
39
9
25'
2 , 913
9 [
350 28
ij+1'
-(1,1)
1050 42' 3
+ 3
1+1,j
(1,1)
15ji+1,:j
1
11 22 1]
1400 t
9
itj-1
1
i+1,j
9 ' 2
P0i+1, j+1
1(0,0)
i j+1j+
i+j+
176
13 13
4200
(0,1)
i+1,j+1
[ ,
+
i
(iii)
-l]
From taking arbitrary variations in
i
=
3
3
13
9
139
l
33
13
T
13
i-ij-1
i-1,j-1
(1,1) (1)
i-1,:j-1
i-2]
1
143 13
2100[42 * 2
(0,0)
1,jt-1
-
-
1(0
1 i-1,j-1
13
+1050[t 42' 3 *
(0 ,1)-
1-1,j-1
1-1,j-1
[- - , 3,
1
1
9
17528s' 4'
+
- 11,-1
1
(1,0)
350[ 28' 2' i1j1i-1,j-1
1 13
14002
1
all
(0,0)
-
i-1 j
4200E
0
(D2.2)
j - 1 to J +1:
2 to I +1,
1 39
350 28'
=
j = 1 to J + 1
1 to I,
=
(3)
P
)
+1)
i,j-1
(1,1)
i-1,j-1 Pitj-1(2
177
P
(-)
13 13
350 12
+(0,0)
-1 +-l
3 13 13
350 4 2' 9
1
143
(1-1, j-1+1-1,j)
13]
2100 42
1
11
+1
+1400
22
21 9
13
13
2
il(0,0)
(i-lj-l+i-lj) ij
-1,j
-1
1,j
1l(01)
1 1122
1050 21 3
11
(1,1)(4
i-1,j-1 i-1,j
1-1,j-
11 9 1
1-1,i
(1,1)
13, 3 ]
+2100 21t
1
(0 ,1)
-1, j-1
[115 22
1 143
1~5 [42
2
i-1,:j
13 1
143
11
P (0 ,1)
i-1,j
+2100 42
p(1,0) (+)
1-1, j
(,1)
i,j
i-1,j
22
1050 21' 3'
i-1,j-1
1(1,1) (3)
178
,j
j 10
6L1
PVT-T ( Z
CT
(04T
P~~ ~
IT- 6 1]OT
M
T
T
(0g0 ai
TT
+
CC
s
I-
?=E
~
T+
-
f6T
W *(T 40) cl TIE uT suoI~vIavA 4AIEI1~qat 2uT)~Ej w~oaa
(c0zxI)
T + f' 04 T
=
i: 'T+ 1 04 Z
=
T+VT
VTE
W (C(IT) al
0
H
TT aVIIT
[TI(-[go)
H T+P
PI
CTCT
Z '4.k 1 Qi7
T4 T-1a VIT[
+OT
C
1u7T TI
CT
VT]-IT
z6
(060)
I+P4 I-I ViT-T
aT6-
I
'E CC
IT
6c77001
I-CI
'I-
CT
I
OV+
(T 0)
W1+4I-T
(0'41)a
aV T-I
T+VI-T V T-I C-4z
(0o10) a
[T
'-
CT -CT
c[
c-I C SE
8 s;
-]--T
(AT)
(0'1)
9 (+)
+ [ 9 -2, 3]i-1j
1
+
139
[05 1,
*2
(i-
1
113 9
p ~(l,0)
P j9
011p(091)
,1]11j1j
1
22
1 1 11
22
1
33
+
(1,1)
9
[2
1
(191) (2)
itj
-1
143
(OO)
]
Pi+1,j
p (1,0)
13]
1
i+1, j
210042
+
[ , -2,
S
-
13
05
+
(+)
(1,o)
j
~
1oso 21 T i-1,j
+17
,j
+1,j)
19i-1,j 4,
+525 14,
15
p(1,1)()
P
(1)
s
,1' 1
11 11
~ 2100ti
2
3 l-]
1 L3(090)
+ 1 [143
175 42 *
1r
210
13
1
-
3)
_
-
P
1)
13
1,
(1,1))
]
(2)ji~ s
P
180
39
1
+350 28,
-13
9
(0,0)
-l
Pi-1,j+1
400
-2'1,j 1 0+1
3 3 13
+ 13
1
[
,
1
13
1
1
[3
,1)
P(0 (1,)
13P1 -,j
13 1(1,1)
9 i-1
1400 It
+
13 1
3 13
_350(429
13
2
1
+
[1
ij+1
11
1400 29
+1
3.9 s 1t
350
-13,
S -, -1]
(0 ,1)
1, j+1
22(1)
9
i-1,j
22
-
ij+1
(1,1)
9
i
9]
]
)
(1,0)
-i j
13
1 T] (1-1,j+i, j)
1400 211
1
))
Pt::
1 143
2100 42
+1
()
i-,j+1
, -1] ( 1 -1 ,j0,j
+21-00
(4)
j+ 1
ij+1
(0,0)
P9i+1,j+
P (1+1
181
(-)
3
-
3
1
(0,1)
[1 , -,
()
P(1)
]
(3)
+1400
j -
+ 1,
i=1to I
(v)
0
=
09
1 to J
From taking arbitrary variations in all P
i
=
1 to I + 1, j
1 39
30 12
-
1-- [
'
-
13
9
2
1143
) (1
i-1,j-1
13
1-1,j 1
(1,0)
13.
+2100 42 * 2
3
-0
-),
(0,0)
,O)j-1
-,-
1
13 3143 193
p(
3 [3429 1
-1](-,j-l+1,)j-)
2100
92 i-i.
j-1
1
(D2.4)
2 to J + 1:
13 - ]P
400 42
1
=
~o~
(1 (0,0)
(+)j
(10)H
-1,j-l
1,j-l
13
s.131
[ ,L 1,
] i-1,j-l+i~j-1)
182
(0,l)
1,j-1
1
11
i
11
22
028
(1,1) (2)
2 ,j-1
3 3-1
350
*28' 1
1
i,j-1
1 1s
+ 140
1
P
(00)
i+1,j-1
-, -ij1 P (1,0)
4200 4
isi-ii+1,j-i(-
10
-5
(1,1)()
39t -13 o- 1 39]9 I
17-
+ 1
is223
(1)-(O
i+1,j -1
13
9 ij-1
33
(1,1) (2)
i+1,j -12
9
[f81 -11,
p(0,0)
]41 -
P-t
1
143 -19131(1,0)
2100 28 9
21 i-tj-1 i-1, j
+
[
1
,
2, 3]
13
113
1050
3
-
-
1 (
P
, 1
[ 11
1]
+
- )
P(1,1)(4
i-,j-
-1,j
](-1,j-1+ij-1)
P
,0)
183
0)
(
2 13
525 [4
1
-
1
-
+1
1
11
3 ,1 3 1
(i-1,j-l+ij-1)
2(,1)
11
33
[E,
9
4 1 ij-1
-11,
143
[4
1050 42
(1,0)
Pi+1,j
(01,1)
1.93
-
ij-1
i+1,j (3) =
i = 1 to I + 1, j
(vi)
(3)
(0,0)
i+1,j
13
, -,
3 x-
*1
P(l9)
22
=
=
1 to I, j = 1 to J:
11 [119
[,1,t1 1]
+
1 [11
0 [1
(0,0)
P(.O
(1,0) )
22
,,
1]1
P
+
184
0
2 to J + 1
From taking arbitrary variations in
i
(0,1)
ij
all
P
(D2.5)
(1)
1
1122
(0,1)
[,
+ 1 14s,-l13]
2100 42 *
1
11
1
1
13
+
P(Qsl)
i +10 i
1 1
(1,1) (2)
i+1,j
13
2'
13
1
11
1400 [19s
151
.
*41 -5iq
1
13
+ 05[t:
1
9j
,- -1]
1
1
1
13 13
14-00 42* 9
(0,0)
1,1+1
(1,0) )
iij+1
22
13
13
420 0 42*t
P (000
i+1,j
i+1,j
1
1
143
21OO002
1050 [4
itj
1*19
, -11]
1
U ) (1)
(1,0)
0
[t
1050
*2
22
14 00[.*
+05
1
(0,1)
ij+1
p (1,1)
3
1i,
*
ij
_
~
j+1
(0,0)
i+1,j+1
(lo)
i+1,j+1
185
-
13
9
13
-1,
1
+
(0,1)
Pi
i+i+-
[ ,1,1]
=0
j = 1 to
1 to I,
(vii)
ilP(3)
J
From taking arbitrary variations in all P
1 to J:
2 to I + 1, j-
-11,
1-[
-
2100 4232
1
11
22
1]
13
13- 0
1
1
1
i(OO)
p(1,0)
,j i-,,1]
021" 9'
-
0=
i-].j
13 1(0,1)
1fP 3 i-~
i
1
1
1050 14
3
11 11
2100 2
1'
i
-1
(+)
P (1,1)
i-1,j
p(0,0)
11 22
1
1050 219 3
'
i-i
1(1,0)
i-
1
11
1050 219
22
3 i-1,j
1575 14
i-1,j
ij
i'j
(0,1)
i9j
(2)
186
+
(D2.6)
itd
13 13
4200 [ts
(0,0)
1-1,j I i-1,j+1
1
13 13
1400 42 9
i-1,j
1400 2 9
1
-1
i
3
143 13
+ 100 [
~125I6 42
1 13
1
11
+ 140021
[
1
1
*
1050
(0)
-1,)j+
Iii-l,i+1
13
(11)
-1
,
00
ii-,
ij+1
1-1,j
*2
13
i, +1
0(10)
22
(0
9,
41, 9J Pi+
19~,~P
1
9 3 1 i-1,j
1 - 2 to I + 1,
1)
(1,1)
i,j+1 (3)
0
j - 1 to J
(viii) From taking arbitrary variations in all P (ll)
i
=
2 to I + 1, j
13 13
+
00[f
-
2 to J + 1:
P (0,0)
-, -1
P-d1+
187
(D2.7)
(3)9
881
(0
4
IIYT
I) l
a-VT-I 1
(O~o)
P'T
-
IIIT
P i-T~ T-PlT-I(T OC
-'T00
I
(11o)
EI
V+iVT T- F'
a
(oI)
T
CI
I
CZ TT+
CTI
I
I
zz
(I'0)
TT cl Ir-PV-T 1
(011)
I-PI
al
(o1o)
Il
.6 cTZ OO#'I
(Z) T--VI
I-VT
(IT)d
I-VT-T-
L-
II
IT
I
.c
cI OOI
1I L
zZIOT
I
CI CI
-4
[I
4T6 I OOZ~
c
I
(+)TPT-Tac I-VT-I 6 s1
C+C
(I" )
Z+I. IO
(I)
I-VT-I c I-Vf4 -T
(I'T )
-
+
1 1
+
22 1(
11
l(0
1)
215i-,j-1
P
i = 2 to I + 1, j
(ix)
(3)
1
143
13
itj-1
1
13 13
1050L42i
1
22
1
+40
13
p(0,1)
1
1 3ij-1
13 13
4200 42
*
i~j-1
13
[2 ,
itj-1
( ,0)
ij-1
j-1
11
1
1
1050
(1,1)
i,j-1
i+1,j-1
(1,0)
, -1]
Pi+1,j-1
1
13 -1 13
-1400[2
+ ~44013
W0
1
i~j
(0,0)
200 42 *2
10
i , j-1
(+)
1 -1
-1)(2)
i+1,j
189
(D2.8)
all P(4)
2 to J + 1:
=
0
2 to J + 1
=
From taking arbitrary variations in
1 to I, j
=
11(00)
11
-10
[
1
, 1,
11
1]9P
(1,0)
22
10d56 21
+
+
1
11
1050 21
22]
3 ij-1
27514$i
1575
1
,jit-1
1
143
-100 [2 , -1
1
11
[
(0,1) H
1,
p (191)
(0,0)
13
]i+l,j
(1,0)
, - s1,
11
]
i
1j-1
=
(-)
3 ±,j-l
1]
1050 14' 3
(4)
t
22
10554E2' -1
(x)
i~j
1,-1
C-)
i
p (l1) (3)
i+1,j
()-
0
1 to I, j - 2 to J + 1
(D2.9)
Zero flux boundary condition on the left:
Set P 0,0)= 60,0)* - 0,
j - 1 to J + 1,
eliminate J + 1 equations from Eq. D2.1.
thiswill
Zero flux
boundary conditions on other edges can be treated in
the same way.
190
(xi)
Zero current boundary conditions on the left:
Set P (
)
P(
(+)* =0
j = 1 to J + 1,
this will eliminate J + 1 equations from Eq. D2.2.
Zero current b. c. on other edges can be treated in the
same way.
191
APPENDIX E
DIFFERENCE EQUATION COEFFICIENTS RESULTING FROM
USE OF THE FINITE ELEMENT SYNTHESIS APPROXIMATION
METHOD IN TWO-DIMENSION
The GxG matrix coefficients resulting from the finite element
synthesis method using bi-linear basis functions in two-dimensional
multigroup diffusion theory are defined below.
In order to simplify the forms of the coefficients, it is
convenient to define the following GxG matrices:
K i,j
y)
=
i*j (x
ij(x,y)h xih
ILX
(x,y) = CiTj (x,y) D-1
IL
(xy) = n T (x,y)Dil1(x,y)hihyj
X
I Y
(x,y) = (
(x
(xy)= n*T (x,y)
(x,y)h
(x,y)h
P
(x,y)h
X
*T
IMis (x,y) = $iq (x,y) cis (x,y)hy
]3p (x.y) = * 9 (x,y) ni
(x,y)hx
192
h
19i (xY)
(
(x,y)
,
(xY)
~(x9y) E
$
R (x~y)-
(x,y-
(x~y) h
xy
(xy)
iDj(xY) h
y(x~y)
yj
hh
()=x
S
SR
(X,)
(x) = h
S
x
*T
for each
=~
*Tregion
)ID
(.)(i,j), and
(x,1)
ID
(x,0)
XY
*1 (x,0)
yj
s
(x) -i*
(x,0) m
(x,0)
i
-(x,1)
.yj
S
(x)
S
(x)
h = j $j+1
*sl
h
=
xi
*T (xl)ID
(x,1)
$i,j+1Xm
yj
for each horizontal region interface and
h
hxi
Ti(Y)
T~i~j
=
hxi
Ty
t
9
*T (O,y) ID
(1y)$
iiji'
'
(l,y)1D
(1Y)
t
$pi+ 1,(j
(ly)
qY)
for each vertical region interface.
193
as
Also define the GxG matrix E ipj (a,b; cd)
1E
(ab; c,d) -
dy <p(
dx
-[ 4 (x,y)]pcx
bX
j
d
db(Y)'
c (x)L(Y)
.(x,y)pc
d
+ qa
b
c+
+ 3R
(~~c
d
+ pax
b
c
+ IR
(~~c
IN
-
cjx1)P(xP~
q())(E.l)
where a,b,c and d are equal to either 1 or 2 and
2
q2
pc(x)
=
(Xp
I
+(1-xP
x
p2p
-
d
=p(x)
-j
1 q
for each region (i,j), i = 1 to I, j - 1 to J. Then with the understanding that all nuclear constants outside the reactor region should
be set equal to zero, we can write the matrix coefficients in the
following form:
194
(i)
conditions
Zero-current or Symmetry Boundar
+
(2,2; 1,1) I(00
-
* -1 j-1)
(1,
1)
it{
-
(y*
fd p (
-
12
+
$T-
on All Four Boundaries
- $ l (0,1)Ti-
(-,j190
Y*
(1,1)T
j-1
-1(0,0)}
*
1-
(00
+
$
T- (11)si-1,3-1-(X l -1 ,
-
--
i(1,0)S
t-+1
i = 2 to I + 1,
195
,3-2(0,0)1
j :- 2 to J + 1
ab
T
T
=
+'ir'9
+
.1x-
0
i1j122_12*_
1i
i
-1
-j
dylYp()VT-1(9)I
1
(0,0
j(0 ,Y N -
00
(190)
Tiiji-1- ij(yW_1
:L-2 j
+
T -1 (0,0,)T
-
-
}
(0, 0)
+ fo dyp 2 (y)p2 (y){i- 1 j ..{ll
x
+I
-1I
i1j
1j -L1j1op)
*T_1 1 3,1) T(Y~v-1(1
+
1.1 lY
R1
- + i-1
196
)
(x)]
1
*T
j-lto
i = 2 to I +1,
a-1
,2
19)
j(2,1;
1i, 0) E
1 to v +
1,-
[Si-1,+j-1
+2(1,o0)
f
-1
9 (0,1)
1
+ I
dypl(y)p2
(Oiy)4
- IR
(Rx
(1 , 0)[
- 1 ,j
1
J+
1,Y)
(0,1)
1
-1
-2,
-1
*-
(00)T
-
-1
1
(1,0)[R x
-1,
+
-R
(
(,1)]$
*T 1 (1,0)
S iI.
+ Ili1-19 j
+ d
-
j
1i,
(0 0)
-1,
(1,1)S-1 v
i = 2 to I + 1,
197
j = 1 to J
,0)
ba,
-1
-I*
(01) E
+ *T
+g-
-T
1211)1
*T l
dyp 2(Y) -v1 (Y)
2 fo
i
v-
00
(1,91)IT()
v-1
T1
+
dxp (x)p (x{4i J-1(Gl
+11
2f0
-
(10
(1)E29;1V-1l
F,
.(x9
Ri
1(,1
0) V- 1 .1 (00)
1ij
pij 1(000)s~~..
-
-1
W isj -l(Ot 0) 1
-
(1
1
(01
(O91)Si~-l
+ p*T
1 (X, 1)
DRy 1
:L- 9j-
IR dx(x)p
x){1
i.l2
22 T 11-1,j-1(1,)
+
-1
11
ix
198
-1,j-
i
bb bb
=
-
1 to I + 1, j - 2 to J + 1
TIP19 (0,0) F
Ei9
*Tl(1,0) E
+
(2,1; 2,1)$p~
(1,0)
(1) E(2, 2 ;2,2)$
+
+ lp*
-1 (.0 0)
9
(1
(1,1;11)l
(1 , 2;1, 2)$p-
1(0,1) E
(1,1)
(0,1)
-1
+
dyp ()p
f
21Jo Y 1 Y/ 1
- T (y)J$
+ $
T(00)
YLj
[T+-
()
~ +~1
(1,0)
(1,0)[T_
(y) -T
j
*T~
dyp2
+f
+
~-
1-1jl,)[
2
Y2 Y(Y)Y){IP(0 1)~
2 Ti-l+,j-1
iij-l
+*T 1-
+
2 odxpy 1( p (){$
(y)-1
j-(0,1) [S ir
199
(x)
(0,0)
Wj-l+
- S
$* (0,0) [S
+
iqj
ax2
+
-
-1
it+J -1
-S
ij-1
i~j-
1-1 j-i1,)[
2
S i-1,j-(xl - 1 ,0)
i+ *T1,
)
S
(9)SW
+
)[i-1+j-1
-
-1,j
-1,j
i = 1 to I + 1, j = 1 to J + 1
*T i
2-- 1 j(lO) E
-,j
(2,1;2,2)$
-1
+ $ (0,0) 1,(1
L dyp
+
-T
+
-
(Y)
p
2 (Y)%
(Y)
- 1 ( 1 , o) [T-
y
R
(1,1)
(0,0) [T(
(11)
(Y)-T+i
.91(X91)] (0,11)20
1 Y) ,(01))
T
-
2
*(X)47
(09T1(l)s.
f:i
2
-2
(x11-1
IRy
:L-1L9j 9(x,1)-
-19
(091)
i-ij
'-q
11
-l+
i g+
i=1 to I + 1, j
1~-1111+
-1
x~j
-
4.)*T
to J
=1
iq
dyp (y)p (y){ip T
(0911)[E x
i-1
'9)T~
(Yv-
201
(9)
.. (09y)
+
o0
- I
dxp(xp2 (
{V
lxp(
s-(x,0)]$is-
-(0,1)
(1,0)
-1jl(0,1)Sijl()
+
-1j-2
-+-1
.
0,0
3iJ
+
19J
f0
(1,
(12;212)
+
- R
.
I
0)
(01)
-1
(0,0O)(R X(0,y)- R X-(1,y)1
1
~
(1,0)T_ 11(y)
+Ip-
1; 2, 1)$ql (1,
dypYp
+ 4 )ji (0,0)T
j = 2 to J + 1
1 to I,
-1 (0,1)
+
1
-1
i
cb
_(x, 1)
1[F
(y)$
(0,0)
,S(1,0)}
dyp(Y)p 2p(Y){2*T
x (1,y)]$ -
(0.1) [ 1,,_ 1(0-Y)
(1,1)
202
$(1,0)
1
*T
-1
T
-l
*T-1
+7
0
W p2
dxp
+(,
[i+j-1
-
i = 1 to I,
(0,0) E i
cc
dyp*T
- IR
*
-
+
-
-
-
j = 1 to J + 1
(1,1)
(11;22)$j
X
(091 )s
(Y) -1
(0,1)
i+l qj
+ IRj+(x1 ,)
(0,0)
-j() ii1 (10
91
l(1,1) +
(x)
i+si
(1,1)}
(Y)$
dxp1 (x)p2 (x)lpj
K, (x,1l$
01R~(O9y)
T'p
(1,1) + $ :iq
j (OsO)T
(1,0)T_
I
-1
1 (Y)P2(0Y,0)11
(1,y)]
(1,0)
()Sj1
j-1(0,91) { S.
1
i = 1
to I, j
203
=
1 to J
(ii)
Zero Flux Boundary Condition on Certain Boundaries
If the left boundary has a zero-flux condition and the
other three boundaries have symmetry conditions, then we should use
the following equations:
bb
ba
-2,
2
2,
1 + bc
P
+ cb
+-2,1 2,2 +-2
P
+cEC
P
P 3,1
-2,1 3,2
,1
P
P
+bb
j 2, j-1 -2Pj j
+ca
P
P2, j+l -2,j
+bc
-2
P
0
P
+cb
3,j-1 z2,j
P
=0
+cc
3,j -2,j
3,j+1
j = 2 to J
ab
P
ba
-2,J+l
-0
P
+b
+cP
P
+ bb
+ -2,J+l 2,J+l + -2,J+l 3,J + -2, J+l 3,J+l
P
P
+ac
+bb
P
+bc
0;
+cc
P
z-i 91 i+l ,2
aj P
+ bc
+
+ab
P
+ca
Pi1i,j+1
P
+ ac.
P
+cb
i
P
P
+ cb
P
,j i+lj
P.-4,
i+l1,j-1 +
P
3 to I + 1
+ba
+c
+--,
P
+bb
P
Pi+lj+1 = 0
i = 3 to I + 1
j = 2 to J
aa
P
+ ab
P
+ ba
P
+ bb
P
-,J+1 i-1,J
i,J+1 i-1,J+1 +
i,J+l i,J
-i,J+l 1,J+1
+ cai,J+1 i+1,J +
i,J+1 i+,J+1 = 0;
204
i = 3 to I + 1
P
where the matrix coefficients are defined the same way as in
The
(i).
equations and coefficients for zero-flux conditions on other .boundary
or boundaries can be found similarly.
The implied zero-flux boundary conditions, as defined by trial
functions Equations 4.11, will result in equations having the same forms
as ordinary zero-flux conditions.
However, the matrix coefficients
must be changed by putting the appropriate modified basis functions,
similar to Equations 4.12, in the corresponding regions.
205
APPENDIX F
LISTING OF THE COMPUTER PROGRAMS
FORTRAN listings of programs LIN1,
LIN2, DOB1,
DOB2, MAN1,
and MAN2 are listed in only the first five copies of this report in
the following six sections.
206
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